This is a tough situation for you to be in.
Some thoughts:
First obviously you don't have much time to prepare, even if the group practices with you, so you need to prepare ahead of time somehow by yourself.
If you have never heard those songs before, it will be very tough to hang in there with them after only a few practices.
One aspect of playing along with others is the distraction/oddness. You hear more notes and rhythms than your mind is accustomed to, so it takes more concentration to 'listen to yourself'. It's easy to get lost, especially the first bunch of times. Also they may have breaks, stops, volume/tempo changes, and other things that are comfortable to them, but just another 'difference' for you.
To help with this, maybe try playing along with simple songs as Eddie says, from Youtube or some other source such as cable/satellite
radio if you have that. The only problem there is figuring out the keys so you can play along.
Second, for this instance, I recommend a minimalistic approach. Hone in on just playing the chords on the beat and strumming the proper rhythm. Sometimes I'll just play the 1 and 3 count's chords if I'm not sure what rhythm the rest of the group is going to go with.
If you can figure out the chords from the sheet music, write them on a sheet of paper. Write them above the words, I presume there are vocal parts.
Sheet music has symbols for 'go back to the top', 'repeat this verse', 'go back to somewhere else', 'skip ahead to a coda'. You need to know this stuff too. Wikipedia is a source for musical symbols.
Piano players can turn the pages on sheet music somewhat easily, not that easy for a guitar player. One way to get around this is to have a helper do that.
To your question about how the key fits in - piano 'key' and guitar 'key' are the same. If they said it's in 'A' for the piano then it's in 'A' for you. You can tell by the 'key signature' which is in front of the clef symbol (looks like a real fancy 'G').
There will be 0 through 5 sharps (#) or 5 flats (b) there.
Key of C = 0 #/b, 1,2,3,4,5,6 sharps mean the key is G,D,A,E,B,F# respectively, 1,2,3,4,5,6 flats means the key is F,Bb, Eb, Ab, Db, Gb respectively.
I presume you know what it mean to play the chords to a song in guitar in a certain key.
While you're playing if you get lost, just stop and wait until you can figure out where you are. In this situation playing the wrong chords is worse than playing none at all.
Sometimes I'll just play the bass notes (1 and 3) if I can 'hear' them until I get back in the game.
About turning down the offer, I'd say it's better to do that, than make the group dislike you because you played and it didn't go well.
What I say when asked to play with a band (more than just sitting in at jam sessions) is that I have to work on my own songs and just don't have time to add others songs to my repertoire, and also it isn't proper to ask them to wait for me to figure out their songs, or for them to figure out mine. I also say that I have limited practice time and need to stay on my own songs at those times.
Now I'm not you, so the next paragraph is subjective. If I were you,
my inclination for this situation is to decline this one, citing the short period of time. If you/they are interested in a longer-term gig, then you can start practicing and see how it works out.
To smooth it over you could go to a practice and see what they expect and how close you come to it, and then if it is a bad fit, decline. I'd assess this as that there is a larger probability of a bad experience than a good one, given the circumstances.
Don't like to be negative, but hope that sounds realistic to you.