Transport: Kobe to Arima Onsen and to Osaka

Could you please let me know how to take the express bus from Kobe Sannomiya to Arima Onsen.

I understand that the Hankyu and Shinki Buses go from Sannomiya to Shin-Kobe to Arima Onsen, so it takes longer, compared to the JR Bus, which goes from Sannomiya directly to Arima Onsen. Is that correct?

Where can I buy the bus tickets? Do I need to buy them in advance? I am planning to reach Arima Onsen at around 3pm (after lunch and at about the time when my hotel allows checking in). Is the bus ticket 700 yen per person, or is there any cheaper ticket for children age 12 and elderly above age 70?

The next day, I would like to travel from Arima Onsen to Osaka. What mode of transport should I take? My group consists of a family of 4 adults (including 1 elderly) and 1 child and we will have a few pieces of luggage. Is it faster/cheaper / more convenient to take the train or the bus? My destination in Osaka would be near Tennoji station. We'll be traveling in early December.

I would be grateful for any advice. Thanks.

I understand that the Hankyu and Shinki Buses go from Sannomiya to Shin-Kobe to Arima Onsen, so it takes longer, compared to the JR Bus, which goes from Sannomiya directly to Arima Onsen. Is that correct?

Yes, if you depart from Sannomiya, then the JR highway buses are faster, as explained here:
https://www.japan-guide.com/e/e3558.html#section_get_there

Where can I buy the bus tickets?

At the ticket counter, online, at a convenience store or by phone. I am not aware of an English online reservation site for the Sannomiya line.

Do I need to buy them in advance?

You can buy them at the ticket counter in Sannomiya (Sannomiya Bus Ticket Counter) just before departure, assuming that vacant seats are left. I don't think that these buses usually sell out, except maybe at the beginning of weekends or holiday seasons.

Is the bus ticket 700 yen per person, or is there any cheaper ticket for children age 12 and elderly above age 70?

770 yen is the regular one way fare for the Sannomiya-Arima route. If you reserve at least 7 days in advance, you can get a 70 yen discount. Children aged 6-11 pay half price (rounded up to the next 10 yen), i.e. 390 yen. I don't think there is a senior discount.

The next day, I would like to travel from Arima Onsen to Osaka. What mode of transport should I take? My group consists of a family of 4 adults (including 1 elderly) and 1 child and we will have a few pieces of luggage. Is it faster/cheaper / more convenient to take the train or the bus? My destination in Osaka would be near Tennoji station. We'll be traveling in early December.

Taking the train would involve a lot of transfers. I recommend to take the bus to Osaka and then the JR loop line or subway (or taxi) to your final destination.

Regarding the reservation for the bus from Arima Onsen to Osaka, you may call +81668663147.
Please refer to the website below:
https://www.hankyubus.co.jp/highway/route/info/highway14.php

You need to buy ticket 30 minutes prior to the departure.

One of my days is going from Kobe to Arima Onsen, I will be going from Shinkaichi Station (my hotel is right next to the station) which require you to just switch train at Arimaguchi Station on the next platform right next to each other, if you are coming from Sannomiya Station, you are require to take a 6 minute train to Shinkaichi Station passing through 3 stops.