I was sick in Kyoto and thankfully my hotel staff were there.

No.1 will be most convenient.

You can use Hankyu Line and change at Katsura for Arashiyama.

If you walk to Gion Shijo, you can use Keihan to Inari-mae.

You can use the subway to Kita-Oji and thirteen minute bus ride or ten minute to Kinkakuji Temple.

If you do choose an Airbnb, message the host in advance to gauge their communication and response rate. I have not stayed at a Japanese Airbnb but most Airbnb hosts I have talked to respond quite quickly. You can also ask them for a nearby hospital or clinic in advance. To communicate with the doctor, you might need to use Google Translate (a big reason to have live internet data access all the time).

Most of the cases, I think good rest will cure most of the things. About diarrhea you just need a bathroom close by.

If things you need an ambulance tell the people nearby and they would get one but just in case of emergency, please. It is better to head for a small clinic before you go to a hospital. The wait will be pretty long.