Do many Japanese people speak Korean?

No. Most Japanese people do NOT speak Korean. However, the English language is a required subject in the Japanese secondary education; although English education has not gone very well for Japanese people, in general, most people can understand at least a little bit of English (except, of course, the very old people). (EDIT: As commenters reminded me, there is indeed a increase in the number of young Japanese people who study Korean (in their mean time, or perhaps as a second foreign language in college), due to the popularity of Korean popular culture among the young generation. However, there are not that many of them such that you could randomly run into them on the streets.)

There are places with Korean signs (usually touristy places), but there are far more English signs than there are Korean signs, so there's no real advantage here if you read Korean. In shops frequented by tourists, there are often Korean speaking staff members; but of course, English is almost always more useful. Nevertheless, all of these are almost exclusive to Tokyo, Nagoya and Osaka metropolitan areas ("Tomeihan"), and you should expect almost no Korean signs outside of this area (perhaps some in Fukuoka as well, due to proximity to Busan).

If you are able to write some hanja (Chinese characters), however, you might be able to express simple ideas and communicate with Japanese people using them, because virtually all Japanese people know some kanji. Ability to read Chinese characters can often help you understand signs as well. But do not expect too much out of this.

There are around 1 million or so Japanese Koreans (including N/S Korean citizens living in Japan, and Japanese citizens of Korean descent), but you could not tell them apart from the ethnic Japanese, so it's likely not a good idea to try to speak Korean to random people. However, there are a few Korean ethnic enclaves in Japan (such as ??kubo in Shinjuku Ward, Tokyo), and you'll of course have luck speaking Korean in those areas; nevertheless, none of those Korea-towns seem to be popular tourist destinations.