Well the singing overdubbed will obviously work. I however, would usually do it together. There are good ways to get decent isolation during tracking for just vocal and piano and to me usually their is an interactive quality that is very important in piano vocal setting, but depends on your music style. I doubt any jazz or classical player would go for it, or they would certainly feel out their element. When I do jazz group w/ singer, I either do the vocal tracking simultaneously with make shift methods of isolation or in total isolation from other instruments, but don't overdub as it is just to unnatural to do with this music.
Again, best piano sound will depend on your room, so experiment with placement of the mics, from distant to more close mics. If there is a singer you might need to compromise for closer placement. If the room sound is not acceptable, you are forced to really go the Jazz route and mic close to the hammers as main track. It can also get a very nice sound, but not the natural spread and reverb u might capture, but thats not there to be had if room is not good. You might consider a Gobo style barrier between the singer and piano and one of those stand mounted vocal isolation mounted contraptions. Distance miking has advantages, but none if acoustics bad and that is harder to overcome post process than adding fake reverb to something close miked. Also it will expose the noise floor in cheaper mics and i your preamps, as you will need to add more gain, so if your gear is not up to par it will introduce problems with noise.
Upright try both from the top with the top open and from behind the soundboard, for up close.
I record mostly jazz these days and I do the up close, but also a stereo ribbon maybe 2-3 ft above the open lid. Not distance mics, but I get a bit more body and of course, room, and in case drums, bass, but it sort of functions as a room mic too.
and some light EQ may be in order, depending on your mics, and the qualities of your piano. For example, my ribbons, usually need a slight boost in highs, very slight (typical with ribbons) and my SDCs often need a bit of hi gain reduction but again vey slight. Compression, similar, depends on the setting, if the piano is solo or is playing the bass lines in the piece, very very little if at all. If in a mix with other instruments, I do usually do a little. This is acoustic piano, so keep it natural and try to embrace the upright characteristics instead of EQing to make it sound like a Grand. Some prefer tracking with top uprights. Use your ears too, stand at varying distances/positions, how does it sound, muddy, boomy, distorted, ringing. Hope fully your gear will capture how it naturally sounded and in this case hopefully u will like how it naturally sounded.
