You can't really get lost on Ulleung-do island; while the terrain is spectacularly steep and forested, there's only one coastal road, and it doesn't even completely ring the island yet. When you disembark from the ferry at one of the two tiny ports on this seven-mile-wide volcanic summit, among those greeting you will be rental-car-agents. Don't be dismayed by the quick transaction--give one of them your credit card, and they'll scan it immediately with their smartphone, and you'll be driving in minutes. The agencies really don't worry about you stealing a vehicle--where would you go? Do be prepared for waves crashing over the road as you curve around the spectacular coast; it's often one lane and the seas can be rough. And don't underestimate the driving time--even if the island is only seven miles wide, it can take up to an hour to get from the port to the opposite end of the island.
You might find yourself being one of the only non-Koreans on the island if you go; go now before this 'mysterious island' gets marketed to the outside world: hiking, seafood, and isolation abound.