If you want to give tuning a go (which you clearly do) fine but I would buy a proper tool for this. Maybe you already know all this stuff but anyway, turning the pins on a piano needs a lot of force but in a very controlled way (minute movements) i.e. you need a long rigid lever. I think it would be much more fuss to try and make one that buy one and then you can get to the bit you want to - the actual tuning.
As to the tuning itself. (not to put you off) but it really does take considerable practice (I would say years) to do this properly. But heck, have to start somewhere. Yes you do need a "good ear" too as for some of the intervals in the temperament octave(s) you will be listening for very fast beats. You have to know how to fix false beats and understand enhamonicity enough to allow for it (due to the piano wire rigidity effects).
You'll also need muting felt and rubber wedges, plus strong wrists determination and patience. As I say though, not in the least trying to put you off - quite the opposite - just mention this stuff so be prepared for this non-trivial task.
As to tuning to concert pitch, well if the instrument is a semitone or more flat, I would be careful as the strings can snap when trying to pull them to pitch. If you do try, suggest using a toothpbrush to clean the contact points and around the tuning pins to remove any surface contamination.
Personally to accompany a singer - or any other intrument, I'd like a piano at concert A - don't fancy transposing a score at sight Well I suppose you could try asking the singer to sing flat -a bit (then run)... :-)
hth.
ps I've known a few piano tuners who couldn't tune a