If it was turned OFF when it got wet it may be OK as long as you 100% dry it out before powering it back on again. It all depends on what temperature the water was and if the heat has damaged anything.
However, the battery would have shorted and is most likely no good and probably in a dangerous state. Maybe someone else can offer advice on the battery but I would suggest replacing it.
It depends on how the Arduboy circuit is configured but you MAY be able to power it from USB. If so the current from USB is more than enough for the low power devices on the board.
Try and remove the PCB from the case and then leave it in a bowl of rice for a day or two before you try it out.