The SATS lounges have bathrooms where you can take a shower, just ask the counter staff when you arrive for the towel. Note that ordinarily you need to get a lounge pass at the time you check in. If you get the boarding pass when you arrive at SIN you will get the lounge pass at that time (btw go to the transfer desk in T1 airside to do this). If your boarding pass is issued at your originating airport, I doubt you'll get the lounge pass, but I presume you will be able to enter the lounge regardless.
If you have * Alliance gold status with a Star Alliance airline, note that there is a Thai Airways lounge close by that you can visit, assuming they are open at the hour you arrive.