Can I run a sink drain horizontally through an adjacent cabinet?
The hot and cold lines are a fairly simple matter. What I would do is elbow 90 them and then run CPVC up a bit vertically and attach an inline cutoff (not a faucet cutoff, I'll explain why in a minute). I would attach maybe a 2x3 or 2x4 in the back to strap these to, and then elbow 90 them off again, just below the counter (might not be a bad idea to strap them to the counter for this step but CPVC should be rigid enough for this). Now run them at a 45 degree angle to the wall of the cabinet adjacent to the sink.

Once we reach the wall, cut holes, attach a 45 elbow to each line, then feed the pipes through and 90 elbow them down the wall on the opposite side. Or, if you prefer, you can 90 elbow them to the back, another 90 elbow to the left, and then 90 elbow down (this will require more complex running on your part). Finally, we put a faucet cutoff valve here, where the faucet will attach.

Why two cutoffs? Let's say you're loading something into the cabinet on the right and damage the piping somehow (it's very exposed). The extra cutoff will allow you to shut the pipe off there. Think of it as insurance against replacing the cabinet and anything in it.

The drain is a bit more difficult. Measure to the top of the drain pipe in the wall and take note if they ran 1 1/2" or 1 1/4" to there. The drain will have to take up some room on the back wall and will be in the middle unavoidably.

On your sink side, install the drain and P-trap (buy the whole kit with the wall tube). Now, on the wall adjoining the other cabinet in the back, mark 1/2" above where you measured the top of the drain in the other cabinet. We need the extra height to ensure the water flows down to the drain. Cut a hole here. Put a 45 elbow (if they installed 1 1/2" on the drain, here is a good place to transition from your 1 1/4" drain pipes) and then put your drain line (PVC or ABS) along the back wall. I'd aim for a 1/4" drop here, then 90 elbow and drop the rest of the way to your drain, putting a final 90 elbow here.

It's a lot of piecing and gluing, but it should be out of the way to maximize cabinet space. You should still have post of your cabinet space this way. I would not route them squarely in the middle