I am replacing an old formica backsplash with tile. On a small section, the formica backsplash was glued onto the side of a kitchen cabinet. I have removed this portion and sanded off the glue, now I have basically bare wood. What do I need to do to prep this surface for tile, or can I put it directly on this cabinet?

As a secondary question, The remainder of the formica will need to stay up. (It is darn near impossible to remove without the exposed edge to help pry it off) I am sading off the glossy finish of the formica, is there any other recommended prep that needs to happen? I would prefer to avoid adding cement backer board as my eletrical outlets will need extreme reconfiguring if I add that much additional material. The formica is glued to a medium density MDF that has held up well for nearly 30 years.

I would have left the Formica on and rougher it with 100 grit. Use Acrylpro or similar and you can bond it to the cabinet.

The board was so thick, that I would have to edge it and there was a curved edge that had significant design and technical challenges. This will work on a majority of the backspash where the formica is staying put, but the one piece on the cabinet would have been a nightmare to deal with.

If I understand your situation correctly, you want to put some tile on a wooden cabinet that is at the end of your counter to cover where you remove the back splash.
I would just attach the tile with an adhesive caulk. The caulk will certainly hold and will remain flexible enough for any wood movement in the cabinet. Just put an X of caulk on the back side of the tile. Put the tile in place, move it around a little to spread the caulk and press it flat. We don't always have to make something difficult.