I highly recommend the Miho Museum. It's not a very good use of your rail pass, but it's totally rainproof. If you time your transit carefully, you can have two or even three hours there and still get back to Kyoto Station by 2:15. Access information is on the museum's web site. (Tip: Get to Ishiyama Station early enough to buy a museum+shuttle bus ticket and get in line at the shuttle bus stop so you can be sure to get a seat.) They have a fantastic antiquities collection, amazing architecture, a current exhibition that looks very interesting, and a spectacular natural setting that is beautiful on a rainy day.
I really like the greenhouse at the Kyoto Botanical Gardens. Entry to the gardens is close to the Kitayama subway station. You'd need an umbrella to get to the greenhouse, which is sort of in the middle of the grounds (so a couple of minutes to the entry and then maybe five minutes to get to the greenhouse), but after that it's rainproof.
If you haven't explored the upper levels of Kyoto Station, that could be good for maybe an hour. Between the greenhouse, the station, and lunch somewhere that would fill up most of your time.
There are a number of covered shopping streets (shotengai), and those might be good for a stroll, but my favorite place to buy things in Kyoto is actually the Katsuragawa Mall, which is five minutes from Kyoto Station on JR. It's not for everyone, but there are some very good stores there, and also some good restaurants. Completely rainproof. On a Sunday it might be crowded, but not with foreign tourists.
Maybe soak at a sento. I'm not familiar with the ones in Kyoto but you can find a number of them using a web search. For example, here's one listing:
https://www.japan-experience.com/all-about-japan/kyoto/attractions-and...I have sort of had my eye on the following one, which isn't on the above list for some reason:
https://heianyu.com/If it were me I might take a train ride somewhere that makes good use of the rail pass, but offhand I don't have any particular suggestions. Possibly a loop around Biwako? I often take day trips where the train ride is the main event, and I don't try to pack in much sightseeing. Sometimes I don't do any formal sightseeing. Just watching the scenery go by and having coffee and pastries or a bento can be very enjoyable.
That's all I can think of offhand, but there are lots of other possibilities. (Kyoto is SO much more than "all the major sites.") I recommend going to the Kansai Tourist Information Center in the Kyoto Tower Building and asking for suggestions.