I can't remember properly, but I think if you capture the top region of Kyushu(the island Shimazu start on), then you can get across from there, on a landbridge? You probably haven't captured it yet, since you only have basic ships and that region unlocks a ship. Try capturing that province and see if you can get across from there.
If it is a low money battle(5000), just get as many bow kobayas as you can. Your opponent will probably get a couple of medium bunes, so as long as you stay away and don't let them board, you should win. Medium bunes and bow kobayas have about the same archer damage, but the smaller ones can use flaming arrows, so it is quite easy to win.