Your ship might not have any delta-V left, but that doesn't mean you're totally out of luck; your Kerbal passenger does have some gas left in the tank.
You pretty much have 3 options: either your Kerbal can get out and push, he can say "screw the ship" and just save himself, or you can just wait until the SOI change and mount a rescue mission. There's pros and cons to each.
If you get out and push, the best way to go about it is to set up a maneuver node that requires the smallest delta-V in order to miss Kerbin, point in the direction of the node, then using the RCS backpack, have the Kerbal push from the back. The advantage here is that you can keep the command module, which might be useful down the line, but on the flip side, it takes much more EVA fuel (and time) to succeed, and it's by no means easy.
If you decide to go with the second option, you'll still probably want to create a maneuver node, for two reasons. First is so that you can point the ship in the correct direction, which will give you an indication of which way to thrust when you go on EVA. Second, it'll tell you how much delta-V you'll need to spend. According to the wiki, you'll have about 600m/s to use, which should be more than enough. The biggest problem with this is that you have no nav-ball, and the only real point of reference is the sun. Radial and anti-radial burns are easy, but everything else is nearly impossible.
Your third option, which is basically required for the other two anyways, is to just wait. I have a detailed answer already about what you need to do, but the gist of it is you have to launch your rescue ship along the incoming path of your target, rendezvous, then complete an orbital capture before returning to the surface. As I said in the linked answer, this will take a lot of delta-V to do, since your closing speed could be over 1000m/s. Luckily, it can take a few days for a vessel to go from the edge of Kerbin's SOI to periapsis, so your launch window is pretty big.