Buy train tickets in advance

I use HyperDia and mobile app for planning trips and usually it is a compound trip.
For example:
10:13 - 10:37: Shibuya �� Tokyo station
10:43 - 12:30: Tokyo station �� Choshi
13:05 - 13:17: Choshi �� Ashikajima

And I can't understand how I am able to buy ticket for the second train between 10:37 and 10:43.
Can I buy them all in advance in Tokyo or maybe can I buy them inside the train? Or it is impossible to switch between train so quickly? Please, advise something about such long compound train trips.

If they're all JR, then you can buy them all together at your first train station. You can also take an earlier train to Tokyo Station and buy it there.

The first leg, I would use an IC card.

The second leg, if it is a limited express train, you could by the ticket in advance. Or IC card if this is not available.

The third leg being a different railway, you'll need to buy a ticket once at Choshi Station.

Where exactly is your starting point?

You had better leave Shibuya earlier.
The two JR platforms in Tokyo Station
are apart (both horizontally and vertically) from each other.

The second train is a JR Limited Express train.
In addition to the JR basic fare
for the route Shibuya -- Tokyo -- Choshi,
you pay the Limited Express surcharge
(1340 yen without seat reservation
or 1660 to 2060 yen with your seat reserved)
for the section Tokyo -- Choshi.

You can
(A) use a Suica or PASMO card until Choshi,
assigning the stored e-money for the basic fare (2268 yen),
and purchase the Limited Express ticket on board
or, when you reserve your seat,
(B) purchase both the basic fare ticket (2270 yen)
and the Limited Express reserved seat ticket
in advance at a JR ticket counter.

For the Choshi Electric Railway section Choshi -- Ashikajima, y
ou pay the fare (270 yen) separately.
IC cards cannot be accepted there;
in the case (A),
you touch the card to the reader-writer device on the platform,
which is at the boundary between JR and Choshi Electric Railway.

Which date?
BC it may be different for several mins btw weekday and weekends also Natinal holidays,
and you may be able to use discounted ticket if a day trip or total.
https://www.jreast.co.jp/e/tokyowidepass/

Basically, that plan is hard at Tokyo station,
one or two(some mins) earlier Yamanote line train better if want.
BC "LtdExp SHIOSAI" dep from deep underground Keiyo line platform like this.
https://youtu.be/9gZX4xAzHeQ
(This video recording started next platform of JR Chuo line, though)
Tokyo station map :
http://www.jreast.co.jp/e/stations/e1039.html

So, I recommed take Subway Hanzomon line until Kinshicho,
it will be really better than tranfering at Tokyo.
https://goo.gl/maps/NGCgZp5gtEr
From Subway Kinshicho to JR Kinshicho is not so far, transfering is easy.
https://youtu.be/QhgoXsuYrLU
Yamanote line is no close straight in this case, just doing possible
https://goo.gl/maps/gtKTEdio2uz
and if you want to eat a special train lunch box "ekiben" at Shiosai seat,
you can buy it in the middle of transit without ticket gate out.

If you take Yamanote line first, you can purchase "base fare" until Choshi station
and "Ltd express ticket(reserved,non- or Green)" for taking SHIOSAI from Tokyo station
at JR Shibuya station ticket office at the same time also in advance(*).
http://www.jreast.co.jp/e/ticket/index.html
http://www.jreast.co.jp/e/stations/e808.html
(Automatic ticket vending machines also have English button, but it takes time)

But if you take Hanzomon line first, you can buy base fare and Ltd express ticket
as "from JR Kinshicho to JR Choshi" at JR Shibuya station ticket office in advance,
or you can buy at the window of JR Kinshicho station because you have enough time.
Until Hanzomon, IC card ticket Suica or PASMO faster and better.
https://www.jreast.co.jp/e/pass/suica.html
https://www.pasmo.co.jp/en/

If SHIOSAI from Kinshicho, the Ltd express charge will be a little cheaper,
but back to top, you should find about I(?) can use discount ticket or not.

"LtdExp SHIOSAI" dep from deep underground Keiyo line platform like this

That�hs not correct.
The Shiosai trains depart at tracks in the Sobu Underground annex, not the Keiyo Underground annex.
They make a stop at Kinshicho Station on the Sobu Line.

take Subway Hanzomon line until Kinshicho

I suppose it does not save your time much.

Also the Hanzomon Line platform is deep under the ground.
http://www.tokyometro.jp/station/yardmap_img/_station_%E6%B8%8B%E8%B0%...

And your transfer at Kinshicho is from Tokyo Metro to JR would require at least 5 minutes.
(Personally I have made the transfers many times, through both North and South JR gates.)

Given the Shibuya to Tokyo trains are every two or three minutes it is simple to give more transfer time by getting a train slightly earlier, then the transfer is not a rush. The Hyperdia result is the minimum reasonable transfer - it is not hard to build in an extra few minutes.