Why else take these two pieces from Homer and why deal with Ulysses? Maybe this represents the dead person's character who possibly was a warrior.
Ulysses known for his archery skills is taking on a whole army emerging from the gates of Troy on horseback and in chariots.
The style of the decoration is unique, not so much from an artistic point of view, but for the subject and the colors used.
The style is very simple, it has little to do with later Classical prototypes and rules.
It's an underground tunnel leading to a water supply. Cyprus is full of them. In this case the stairway is slightly deeper, which is probably because people may have not found water at higher levels.