How can I connect to a drain pipe in the adjacent cabinet space?
You'll simply need to bring the drain pipe and supply through the cabinet wall as you would any other wall. Set your cabinet, then cut appropriately-sized holes using a hole saw. Cut from both sides to prevent blowout of the panel's surface. Plan carefully to avoid extra holes, connection problems, and interference with any drawers above.
I would probably use a pair of 45° elbows to bring the drain forward (toward the photographer) from a low point, then 90° through the panel for a trap connection 12-14&quot; off the floor, roughly centered in the depth of the cabinet. Do something similar with the supply.
If you're going to use an air-admittance valve as a vent, that might go well directly above the current pipe location. In that case you'd use a wye instead of the lower 45.
Obviously this disrupts your cabinet storage space, but with a little clever shelving modification you'll have a decent volume.