Welcome to the Japan forum. And best wishes for a successful outcome from your visit to Kyoto.

Which airport are you departing from? That may have an impact on your choice of NRT vs. KIX. I think KIX is worth it if you have a non-stop flight there. If you have to connect through some other airport to get to KIX vs. a non-stop flight to NRT, then NRT is OK.
>> NRT to Kyoto will take about 4 hours with one change of train (about $180).
>> KIX to Kyoto will take about 75 minutes with no changes (about $30).

Is this the medical facility that you want to stay near?
http://www.med.kyoto-u.ac.jp/E/grad_...ess/access.htm

If so, then it's on the East side of town. Hotel choices in the area range from cheap-and-cheerful, Japanese-style guest houses to the Westin Kyoto, which is what you'd expect from a Westin (as long as you avoid the cheapest rooms). If you give a few more details, you might get some suggestsion that fit within your budget and meet your needs.

As for airline choice in economy ... There are numerous discussions of the relative merits of ANA/JAL/AA/UA. A bit more legroom in UA Economy Plus vs. a bit more friendly service in ANA economy ... but they are all sucky ways to spend 12+ hours.