Been there. Done that. I'll try to describe what my experiences have been. I rarely got to practice a accompaniment; usually I'd step in at the last minute for a rehearsal or someone would want to sing and they would ask me to accompany.
You've got a good start by knowing some theory (classical, jazz, etc. are useful.) That means you know the general structure of songs. Since you know the songs, you might just memorize the basic chord structure. It's good to think in terms of (probably simplified) Roman numeral analysis; perhaps just think of major, minor, maybe seventh, diminished chords. The blues progression is good to know as it is very common across many types of songs: I,I,I,I;IV,IV,I,I;V,V,I,I, with perhaps some variation. Often the chords are all sevenths (major-minor). If you are playing piano behind a soloist, there are several ways to play such a pattern: block chords (not really for blues), boom-chick (alternate bass and chords, real good for blues), broken chords (good for varying things now and then), or a counter-melody that basically agrees with the chord progression. You can vary these depending on the text of the song and the desire of the soloist. It's good to write out the progression in Roman numerals (or whatever you are more familiar with) so as to make transpositions easy. Singers often want a piece to be moved higher or lower (depending on the amount of sleep the night before and other matters) or maybe the piano is tuned a half-step low (that's happened in real life.)
There are other common chord progressions (there a free pdf on the internet somewhere listing many such progressions) that you can use: I,I,IV,IV,I,I,V,I and the like.
In your case, probably, you will also play the bass on the piano. A strong bass line can really help. If you look at classical lieder (and other stuff for that matter), the bass and the melody make good two-part counterpoint. A walking bass is used a lot in jazz standards and is useful in connecting parts of a song. Also, you can use a more lively bass line when you have the lead.
If you need to play a solo or break, there are several possibilities. The easiest is to just play an embellished version of the melody. When backing up the soloist, one doesn't want to get in the way, rather to be supportive (mostly play either the melodic tone or the fundamental of the underlying harmony, nothing too dissonant so as not the throw off the singer). When soloing during a break, anything goes as long as you can get back to accompanying at the end of the solo.
If you get some rehearsal time with the soloist, you can work out at least an outline of what you are going to do.
I'm sure others can probably contribute more here.