Kyoto has no airport of its own, so there's really no advantage in flying.

It's 11 minutes by Keikyu train to Shinagawa, where you can hop aboard a westbound Shinkansen. If you are buying individual tickets, any Nozomi (takes a little over two hours to central Kyoto) will do. If you are on a JR Pass, you have to take a Hikari, but even then, it's only about 30 minutes more. (The Kodama stops at the mid-sized cities like Odawara and Hamamatsu and may not run as far as Kyoto, so avoid that.)

If you have a lot of luggage (not recommended for traveling in Japan), take a carry-on with the necessities for overnight and have a luggage forwarding service ship your bags to your Kyoto hotel. They WILL arrive the next day. All Japanese hotels, from the five-star palaces to the tiniest little business hotel, deal with shipped bags all the time and have no trouble holding them for you.