Technically speaking, the Silver Kris and First Class SQ lounges in Singapore are only for their own passengers. Star Alliance members have to go to the "Premier" lounge, which I don't think have showers, and the one open in the morning for the NRT flight is a tiny little place, crowded to the brim.

That being said however, I did get in the other day to the SQ First lounge with a UA FC boarding pass. It is in Terminal 2, so you have to take the first shuttle train in the morning to get there in time for the hearty breakfast if you want to get back to Terminal 1 in time for your flight to NRT.