Asuka just screams "rent a bike". Arrive at the appropriately named Asuka railway station and there will be a bike rental place just on the side. This is the best way to see the area because it is a little far to walk but if you have the time it is also possible. There is an excellent tourist map for the area with many features you are trying to find such as stone sculptures and the burial rocks. If you arrive a little late in the day you can also get the bike cheaper (it was offered by the guy, Japan isn't a place to bargain on prices). Once you cross the main road to the east the paths are too small for cars and you will mainly come across other bikes and pedestrians.

The major place to visit is the Ishibutai Tumulus. There are surprisingly many tourists for something that isn't exactly attractive, has much to look at or is even going to take you more than 5 minutes to explore. I suppose it has a lot to do with the cherry blossom season which makes the area around it beautiful.

Overall you can spend a day here if an event takes place at the nearby multi-purpose rest area but half day is enough especially if you take the bike.