If it ain't broke don't fix it. For five little months, which will go by in a flash while you settle into your house. Leave the old fridge where it sits. Put the new fridge in the garage if it must be deleivered before you complete a remodel. It will give you two fridges to store your stuff in for awhile which is a bonus if you shop at Costco like I do. Your Idea of moving the upper cabinets up is a good one. However its the lower cabinet that will be trouble as it is probably an unfished space below and behind it.

Hey Handyman. Thanks for the advice but I will not have the little frig once I move in. I will be using this in the kitchen either in that location with modifications or on the oposite wall where there is space. I really need a creative solution to the cabinets. Thanks!

www.freecycle.org

Get a new old fridge to use for a few months for free then re-free cycle it.

If you can remover the cover of the radiator or slide it over an inch, you can reposition the right side of the cabinets as well to accomodate the frig. and use a filler piece on the left side to fill in the gap between the side pantry and top cabinet which would not be that noticeable. That would be my suggestion without having to demo the whole thing.

Try to take your top cabinet above your refridgerator apart to se if you have enough space on top and bottom to cut the doors at the top and bottom and also you might have to do that same to the frame around it as well.
Make sure if you need lets say 2" that you cut only 1" from top and bottom.