TL;DR: you'll almost certainly die in the accident, or asphyxiate afterwards.
Will your spacesuit keep you alive long enough to be rescued?
You've got, what, 8 hours max? Maybe a bit more, if you can relax? Unless there's a suitable vehicle immediately ready and able to rescue you that's aware of your plight, you're space junk.
If you weren't in a re-entry trajectory, and you were in a relatively low orbit, your orbit will eventually decay and you will indeed burn up. There's a nifty calculator at lizard-tail.com which suggests that if you are in a circular orbit of 200km altitude or higher, your air will run out before you burn up. By the time you get to ISS altitude, it'll take weeks before you finally get cremated.
Or will your orbit lower to quickly and you end up burning in the atmosphere before someone can save you?
Well now. If you were in a re-entry trajectory, you'd be toast within half an orbit (which depending on where you started could be 22 minutes to many hours). If you were in a stable orbit of any kind, you'd likely remain in that orbit (along with the debris of your former vehicle) until your air ran out or you were rescued. If your vehicle exploded I suppose it might have been able to fling you into a re-entry trajectory without turning you into a cloud of debris too, but it starts looking a bit suspicious that you had time to get yourself suited and booted and strapped into a suitable astronaut propulsion system when the mysterious incident occurred.
According to this blurb on the shuttle's deorbit procedure, the atmospheric entry interface is at ~122km. If you started out at the altitude of the ISS (408km), you can use the vis-viva equation to work out that you need a delta-V of ~83m/s to drop your periapsis to that altitude. This might be excessive; I see a reference to ATV-1 using a 70m/s deorbit burn. In any case, that's quite a clip... 70m/s is 252km/h (or 157mph), and you'd have needed a hefty kick to impart that much speed to you when your vehicle broke up. Accelerating you to those speeds with a bang seems likely to turn you into jam. More gentle explosions won't have enough oomph to deorbit you.
Lets imagine you do somehow end up being deorbited without being pulverised, though.
But then you see a satellite that is travelling relatively the same speed as you nearby so you maneuver over.
(I'm also assuming you and it have coplanar orbits, because this is already fiddly enough without dealing with plane changing)
If you're falling fast enough that you're going to re-enter before your air runs out, and you've come across a satellite that's travelling at roughly the same speed and direction as you, then there's a non-zero chance that satellite has been de-orbited and clinging onto it just makes it easier for observers below to see the point at which you burnt up.
All other satellites can be reasonably expected to be in stable orbits, which means that in order to grab onto them, you'll have to match orbits with them. If you can inject yourself into a stable orbit, then you don't need to grab onto a satellite... you can just do it. Admittedly, having a visible target will make the whole thing slightly less seat-of-the-pants.
The major issue is the delta-V you need to match orbits. As mentioned above, you might be needing 70m/s+ to restore your original orbit, which you'd have to apply as soon as possible. The actually-built-but-never-tested USAF Astronaut Maneuvering Unit had ~76.2m/s worth of juice abord. Every astronaut propulsion system since has had much less.
As for making a rendezvous with a satellite, I'll ignore the implausibility of one just happeneing to be on your trajectory. As you fall down from the ISS orbit at 408km to your atmospheric interface at 122km, lets imagine you meet a satellite in a circular orbit at 250km. As you pass it, it'll have an orbital speed a mere 8.7m/s slower than your own, though it will be travelling in a slightly different direction to you. As anyone who has done manual spacecraft rendezvous in Kerbal Space Program will tell you though, you won't simply be able to brake and catch it easily because you're in a quite different orbit and trying to fly around in orbit like you were in an atmosphere is just not going to work. You first need to raise your periapsis out of the toasty zone to 250km (using up ~36.8m/s of delta-V) and then lower your apoapsis to match, requiring another 45.5m/s, which is pretty close to the delta-V that was needed to deorbit you in the first place and more than you might reasonably find in anything strapped to your back.
(edit: the delta-V would be higher if you fired your rockets once you saw the satellite, because the best and most efficient time to change your periapsis is when you're at your apoapsis, but I'm not going to work out how much less efficient this would be)
The satellite has to maintain it's orbit, so even if you knock it out of orbit a little it should still try to keep it. Right?
I tried to work out what would happen if you just managed to lasso the satellite, pulling the two of you into a new orbit. It is quite beyond my skills to work out actual numbers, but if the satellite is heavy enough you should end up in a stable orbit that's not too dissimilar from the one it is already in. What "heavy enough" means in this circumstance is left as an exercise to the reader.
You'll probably need a nice springy tether to absorb the shock of hooking up to the satellite (it might not kill you, but it won't do you any good at all) and you'll almost certainly end up spinning round and around which might end badly, depending on the rotational inertia of the satellite... if it is too high, you'll wrap your tether around the satellite, speed up and hit it, possibly quite hard. Otherwise you'll end up spinning around your mutual centre of mass like bolas, which you might be able to correct with your propulsion system. The satellite might try to correct its spin by itself if it can (and that's a separate question altogether) which will probably just make life harder for you, as I doubt the designers of the satellite considered the ridiculous situation you've just contrived.
(although I didn't manage to make use of it, I'd like to reference Satellite Splat, a paper on perfectly inelastic collisions in orbit, just because of its name)