The easiest way to make reservations for a ryokan is through a reservation website. While there are dozens of these sites available, the ones mentioned below have some of the best English catalogues and offer competitive rates:
Japanican is the English website of JTB, Japan’s largest travel agency. It has one of the largest English online catalogues of mid to high level accommodations nationwide, including a lot of onsen ryokan, which are hard to reserve elsewhere. Japanican offers competitive rates and powerful search functions for instant online reservations.
Rakuten is Japan’s most popular hotel booking website, and also offers an English version. Their English catalogue is extensive (although not quite as large as their Japanese one) and covers all budgets at highly competitive rates, including a considerable number of ryokan.
Specializing in ryokan, Japanese Guesthouses works like a traditional travel agent, taking orders and then calling the ryokan to assess availability. Therefore, reservations are not instantly confirmed. Their catalogue is smaller than that of other sites, but they have very detailed information on their listings and good coverage of rural areas such as Shirakawa-go, Koyasan and the Kiso Valley where it is difficult to make reservations without Japanese language skills.
Reservations can also be made directly with the ryokan through their websites if they have one. But note that many ryokan do not have English versions, and those that do rarely have an English online reservation system.
Reservations can also be made by phone or fax, but be aware that some establishments may not be able to handle inquiries made in English especially when made over the phone. Alternatively, reservations can be made using a traditional travel agent.
Same day reservations are not a common practice, and many establishments may not be able or willing to accommodate them because food preparations must be made in advance of your arrival. For the same reason, the ryokan should be consulted in advance about any dietary restrictions.
I have also found that some places will not provide rooms for people traveling alone. Some will allow, if the price for two is paid.
Here are some options for Kyoto: