We are arriving in Tokyo on Sat 19 Sept and have to travel straight away by train to Kyoto. As we cannot reserve train seats prior to arriving in Japan; does anyone know if we may have a problem getting on the trains, as the 19th is still part of the Japanese school vacation! (I have visions of white gloved people pushing us and and our cases onto the train!!, not that I mind as long as we can get on!!!) Thanks in advance for any help.

September 19th is a regular school day in Japan. Only universities are on break through September.

Train to Kyoto from Narita will involve at least one change of trains, so be sure you've either packed light or sent your luggage to Kyoto from the airport.

When you buy your tickets you can request seat reservations. If there are none left, you can still ride the non-reserved cars, or you can have the clerk check different trains for you.

You shouldn't have any problem at all. You'll get a reserved seat on the Narita Express from the airport to Tokyo or Shinagawa where you will transfer to a shinkansen to Kyoto. As Kim mentioned you would get seat reservations for the shinkansen when you get your tickets at the Narita Airport station.

The vision that you have is of a Tokyo subway at rush hour.

Exactly, you have no reason to fear being packed into a Shinkansen. It just doesn't happen. I took one (actually it wasn't technically a Shinkansen-too many stops) to Matsumoto once. I was new to Japan and didn't get a reserved seat. I spent a lot of time leaning against a metal bar at an observation window. It wasn't bad, but I always get reserved seats now. Just more comfortable. Expect to be just as comfortable as you would be on a plane but with more leg room. It looks like a long distance on a map but it is a surprisingly fast trip from Tokyo to Kyoto.

jak1975, you definitely were not on a shinkansen to Matsumoto. I am guessing that it was a limited express train (Azusa or Shinano). I would be surprised if you were forced to stand during the trip. Did you look for a seat in other cars or did you just choose to stand?

It wasn't a shinkansen. It made too many stops. I think technically it was a limited express but it was extremely rapid. It didn't fly through the stations like a bullet train but it would have been hard to confuse it with a typical limited express on a line like say the Keikyu line. Ticket cost was around $120 dollars. We had to stand on the way there. We found out that we only had to pay a few yen extra to get a seat and I think we either did this on the way back or at somepoint on the way there. Once again, it's been a few years since I made the trip. It was pretty crowded at first, but the area where we were standing was a small observation area with large bars to lean on and tables. So it wasn't anything like standing on a local train. At the time I was also a smoker so that might explain the reason that most seats were full on the car I was on. Perhaps if I had been on a nonsmoking car there would have been more seats....

There are both reserved and non-reserved seat cars on trains. And both smoking and non-smoking cars as well. The smoking non-reserved seat cars are often the least crowded. The non-smoking reserved cars sell out first. The non-smoking non-reserved cars are often chuck full, with the smoking cars empty. In the non-reserved cars you can stand anywhere you like including the aisle. In the reserved cars, passengers without reservations are only allowed to stand when the entire train is well over seating capacity.

Yeah, the only reason I brought that up is because it's the only time I've had to stand on a train like that. I realize that's not even a shinkansen but it's kind of similar. I know my first time to Japan I thought the Narita Express was a bullet train. Silly now, but it seemed logical at the time. I couldn't figure out why it was going so slow... The only bad experiance I've heard of on the actual shinkansens was some co-workers of mine missed their train. They had reserved seats but apparently since they missed their train they missed their seats too. They could take the trains after but they didn't have reserved seats. Of course it's their fault for missing their train.
On another note traveling from Tokyo to Kyoto means you should get some spectacular views of Fuji, so you may want to have a camera handy. I think it'll be on the right side, but I'm not sure. Maybe someone else can confirm that.