Hakone to Kyoto

I will be travelling from Hakone Yumoto to Kyoto by train on 22nd May this year. I have consulted Hyperdia and the best train to suit my needs leaves Hakone at 9.39. Is it possible to book tickets now and if so how do I do this? Or is it just best to wait until I am in Hakone?

You can only book tickets/seat one month out.

Hakone-Yumato to Odawara will only be a regular commuter, so no need for booking a ticket.

Odawara to Kyoto will be on a shinkansen, so you can decide if you need a reserved or unreserved seat. You can book these at almost all JR stations across the country.

And if you are a large group and/or just want to be safe, book the seat reservation when you first arrive in Japan. It doesn't need to be booked in Odawara, it can be booked anywhere offering JR seat reservations.

I accidentally messed up my seat reservations from Odawara to Kyoto in June 2015 thinking I should try to take an earlier train that was slower. All the reserved seats on my original train were then gone, but I had no problem getting two non-reserved seats next to one another.

Hope this helps!

I recommend you don't rely on HyperDia for train time data in May 2018 at least until April.

JR companies will revise train timetables on 17 March 2018, which includes a minor change in departure and arrival time on Tokaido Shinkansen (Tokyo -- Nagoya -- Kyoto -- Shin-Osaka).
The new timetables are not yet available online.
So, currently route-finding tools (HyperDia, Jorudan Japan Transit Planner, etc.) are not ready for time information in May 2018.
HyperDia tends to be late in taking new time data.

from Hakone-Yumoto to Kyoto by train on 22nd May this year.

What route are you thinking of? From Odawara by Tokaido Shinkansen?

I suppose Shinkansen train seats will not get fully reserved quickly for 22 May 2018 from Odawara to Kyoto.
That day will be a normal weekday, not during a busy tourist season.
Odawara Station is skipped by Shinkansen Nozomi trains, which connect Tokyo and western Japan faster than Hikari and Kodama trains.

How is your whole itinerary?
Which airport(s) will you fly into and fly out from?

Thank you so much for this valuable information.
For interest, we are flying into Tokyo and out of Osaka.

Kind regards