Apartments in Kyoto and Japan are easy get ahold of when you find a "gaijin (foreign) friendly" agency or landlord. Otherwise, it can be a bit of a hassle.
First off, language can be a barrier. If you don't know the language it's a struggle to find out how much your rent is, which money you hand over is refundable and which is a "present" to your future landlord. Also, many normal Japanese establishments require a LOCAL guarantor, which is the equivalent to a cosigner. "Gaijin friendly" agencies/landlords usually speak English, don't require key money (a non-refundable "gift", which is usually in the amount of one month's rent), and don't require a guarantor (or allow someone you know from your home country to be a cosigner).
There are many guesthouses in Kyoto, which typically are shared, month-to-month places. These are a good option and a great way to meet people, however if you need privacy, try to find a agency/owner catering to expats and foreigners.