Biei and Furano day trip from Sapporo

I will be in Sapporo in August. I might have one free day. Is it possible/worth it to see Biei and Furano as a day trip from Sapporo?

I would have to take the JR Lilac Kamui for Asahikawa at 6:30 AM and then take the Furano Line to Biei and Furano.

If yes, is it possible to purchase the tickets for the JR Lilac Kamui last minute? Or will the seats all be booked? Like I said, the free day is not 100%. I don't want to purchase the ticket ahead of time and not use it.

Lilac No.1 has non-reserved seats. Anyone could ride on it. Just decide whether you go at the last minute.
https://world.jorudan.co.jp/mln/en/?p=0&from=Sapporo%28Hokkaido%29&to=...
https://world.jorudan.co.jp/mln/en/?p=0&from=Sapporo%28Hokkaido%29&to=...

Furano and Biei offer a lot. Check their web sites beforehand.
http://www.furanotourism.com/en/
http://www.biei-hokkaido.jp/en/

If your aim is just for lavenders, August is not your choice. Early August could be OK, though, if there were no rains from the end of July.

I will be there first week of August.
The lavender fields won't be in bloom?
Ahhh!
My friend said the flower fields, especially lavender is beautiful.
There are still flower fields though, right? Even if no lavender?
Is it worth a trip from Sapporo?
Seems expensive for one day.

Lavenders should depend on yearly conditions. On average, the peak of later flowering varieties would be around 20 July. One of such, Hanamoiwa, would be good enough around 25 July only at Farm Tomita. It might be better to visit lavender fields at higher elevations around Rokugo Observatory at the southeast of Furano. It is said lavenders there should be still good enough until early August. There are some buses from JR Furano to Rokugo (ex; Furano 8:15-8:55 Rokugo/˜[‹½) or take a tour by JR.
http://www.furanobus.jp/rosen/index.html
http://www2.jrhokkaido.co.jp/global/english/travel/pdf/fb04.pdf
http://www2.jrhokkaido.co.jp/global/english/travel/index.html

Other than lavenders, there should be many flowers, such as in Shikisai no Oka in Biei.
http://www.shikisainooka.jp/green/calendar/

Here are some economical ways to visit Biei and Furano.

1) 5180 yen: Bus + JR (without express trains) + Bus
1340 yen * 2: Bus: Sapporo-Takikawa
2500 yen: JR: Lavender One Day Ticket (Lavender Free Pass: 1 Day):
Takikawa-Furano-Ikutora-Asahikawa by 1 Day Pass and Sapporo-Takikawa by buses. You need to transfer twice from bus to JR at Takikawa.
http://www.hokkaidox.com/wp-content/uploads/2017/05/2-14-BUS-TIME-TABL...
http://www2.jrhokkaido.co.jp/global/english/travel/pdf/fb02.pdf
https://www.jrhokkaido.co.jp/press/2017/170427-2.pdf

2) 5640 yen: Bus + JR/Bus
5000 yen: Inter City Bus Pass (3 Days):
640 yen: Furano-Biei by JR or bus (the same price).
Sapporo-Asahikawa-Biei-Blue Pond and Sapporo-Furano by the Bus Pass and Furano-Biei by JR or bus. If you want to stop between Furano and Biei, you need pay each time. The Bus Pass could be used another 2 days.
http://www.hokkaidox.com/en/hokkaido-bus-pass/
http://www.hokkaidox.com/wp-content/uploads/2017/05/2-5-BUS-TIME-TABLE...
http://www.hokkaidox.com/wp-content/uploads/2017/05/2-17-BUS-TIME-TABL...
http://www.hokkaidox.com/wp-content/uploads/2017/05/2-6-BUS-TIME-TABLE...

3) 5780 yen: JR (without express trains)
1640 yen * 2: JR: Sapporo-Takikawa
2500 yen: JR: Lavender One Day Ticket (Lavender Free Pass: 1 Day):
The JR only version of #1. Takikawa-Furano-Ikutora-Asahikawa by 1 Day Pass and Sapporo-Takikawa by JR. You do not need to get off at Takikawa.

4) 6500 yen: JR (with express trains): Furano/Biei Rail Ticket (Furano/Biei Free Kippu):
The same with #3 but includes express trains with one ticket.
http://www2.jrhokkaido.co.jp/global/english/travel/pdf/fb02.pdf
https://www.jrhokkaido.co.jp/press/2017/170427-2.pdf

5) 16500 yen: JR (with express trains): Hokkaido Rail Pass (3 Days)
All Hokkaido within 3 Days.
http://www2.jrhokkaido.co.jp/global/english/railpass/rail.html

Other economical day trip candidates from Sapporo in early August.

A) 3700 yen:
Sunflowers in Hokuryu would be in full bloom in early August. The past records of full blooms are
1-13 Aug 2016
2-10 Aug 2015
23 Jul - 9 Aug 2014
28 Jul - 14 Aug 2013
4 Aug - 15 Aug 2012
http://portal.hokuryu.info/himawari
1850 yen from Sapporo to Hokuryu by Chuo Bus Rumoi line. ex) Sapporo 720-922 Hokuryo 1647-1850 Sapporo
http://www.hokkaidox.com/wp-content/uploads/2017/05/2-10-BUS-TIME-TABL...

B1) 4320 yen:
2160 yenSapporo 915-1228 Cape Kamui
2160 Cape Kamui 1343-1658 Sapporo
Shakotan Blue up to Cape Kamui.
http://www.hokkaidox.com/wp-content/uploads/2017/05/2-11-BUS-TIME-TABL...

B2) 5570 yen
Go around Sakotan Penuisula.
1900 yen Sapporo 705-938 Iwanai
1510 Iwanai 950-1128 Cape Kamui
2160 Cape Kamui 1343-1658 Sapporo
http://www.chuo-bus.co.jp/city_route/course/common/images/pdf/otaru_08...
http://www.chuo-bus.co.jp/city_route/course/common/images/pdf/otaru_08...
http://www.chuo-bus.co.jp/city_route/course/common/images/pdf/otaru_07...

THANK YOU SO MUCH!!! :)

I read Patchwork and Panorama Road in Biei is very good if you have car...
I don't have a car.

I read there are rental bicycles.
Do you think it's possible to do with limited time?

Bicycles are better than cars in Biei. By bicycle, you could stop anytime you want to stop. But you could not do that by car. You have to find a place to park before you stop.

However, Biei is quite hilly. An electrically assisted bicycle should be a must. Check the year of the manufacture and the amount of the battery. Do not pick a cheap one with an old bicycle with a small battery.

Even with an assist by a battery, there should be a range of time to need to go around Biei by the physical ability of the rider.

One of the leading rental bicycle shops estimated the average time needed to enjoy Biei with their 15 years of operation. They say you would need about 7 hours to do both Panorama road and Patchwork no Michi. The total length starting from JR Bibaushi could be about 50 km. So, the battery requirement for this will be 10Ah.
http://www.yamagoya.jp/renta/

Here is a map for bicycling in Biei.
https://www.biei-hokkaido.jp/cycling_map/images/biei_cycling_map_all.p...
Altitude graphs are shown for Panorama Road.
https://www.biei-hokkaido.jp/cycling_map/images/biei_cycling_map_panor...
The one for Patchwork no Michi.
https://www.biei-hokkaido.jp/cycling_map/images/biei_cycling_map_patch...

Please try to translate the following, one the complete articles on the bicycling in Biei.
http://www.yamagoya.jp/renta/info2.htm

Luisjp gave a lot of ideas. I did not examine it carefully, but I don't think I saw this one:
http://www2.jrhokkaido.co.jp/global/english/pticket/pdf/direction04.pd...

It is a very economical ticket (2260 yen), which gives you a day of unlimited train use, for weekend and holiday days only. You must buy it the day you use it. Only valid on regular and rapid (not express/ltd. exp) trains.

In addition to Sapporo, Furano and Biei, it can be used to a number of other destinations.

I'd check Hyperdia, though, to see how long those slower trains take, and how often they run.