Removing Nicotine Odour from Walls Behind Cabinets?

This is a subject that interests me because I am allergic to nicotine smoke. Unfortunately it's quite difficult to remove Nicotine odor from walls short of removing and replacing the drywall. I have found that there are odor neutralizing components in simple household baking soda(you can purchase it in a bag in the laundry section at Walmart & other grocery or hardware stores). I scrub with it & then put a little in a screened container/shaker in the cupboard.

Borax is also odor neutralizing. I have also used borax to wipe down walls etc., but it is not lasting. Another thing that will continually work at absorbing odor is charcoal. I put a hand full of grilling charcoal cubes in a small open container in a back corner of a closet, cupboard, or whatever. If you want to go to a bit more work, KILZ paint also works relatively well at sealing away stains and odors from nicotine smoke. Good luck!

It sounds like this kitchen may be a "make-over" and the old cabinets were removed and new cabinets installed without doing a proper cleanup. Which probably also means the workers were smokers and could not smell or care about the odor (or just ignored it or did not include clean-up in their contract).

You must like your apartment or it seems you would just move. This will be a very time consuming job so be sure you know this before starting and have all supplies ready before you begin.
One word of caution - you have not mentioned upper cabinets. If they are also "odor infected" then you should do those before doing the lower cabinets. You probably should clean the underside of any upper cabinets first - even if you do not clean the insides.
I would suggest that you remove everything that you can from your cabinets and remove drawers as well so that you can reach as much wood/wall as possible with a spray liquid. Place everything someplace that you can leave it for a day or so.
Buy one or two spray bottles (Dollar Tree will work) and some household ammonia. I believe the lemon scent ammonia will work. Also, buy a pair or two of rubber gloves (like for dish washing), a package of face masks, and a bucket. Several rags for wiping. Of course you may already have some of this.
Very important!!! You must have good ventilation so open every window you can and turn fans and AC to full power. You should not breathe the fumes. This may sound really dangerous but it can only be dangerous if you do not follow just basic rules when using something that creates fumes.
Since this is a tough job, I would make a mix of 1/2 ammonia with 1/2 water in each bottle. You could use it stronger but this may be good enough. Fill your bucket with fresh/clean water and have several cleaning rags available nearby (some for rinsing and some for drying).
Hopefully your spray bottles will have stream or condensed settings as it would be best to not use a wide spray. Be very careful and spray every area inside your cabinets (tops, sides, especially any cracks, so that it literally runs downs into every crevice. It may run to the floor if there is an open area but that should not be a problem as there is probably an odor there also (just forget about any run down unless it runs to outside the cabinet). Just leave this for a few minutes (15 maybe). You can also spray the doors but place lots of newspaper or garbage bags under the doors to catch any dripping from hitting your floor. Wipe your doors with clean water - several times.
You can now get fresh clean water and rags to wipe inside your cabinets. Just wipe every surface you can reach. It will not matter that you cannot remove every trace of the ammonia as it will not hurt you later. You can also use a clean spray bottle for this and just wipe the surface after spraying clean water.
Your drawer area will have to be done the same way even though the space is small. Spray cleaner inside and wait - then spray clean water and wipe as much as possible.
Your cabinet drawers will also have to be cleaned. This should be done outside if possible - if not - then try to do this cleaning on newspaper or plastic on the floor. (Some would say use the bathtub but I do not believe you can get good enough ventilation in a bathroom.) Spray the inside and outside of each drawer and wipe down with clean water.
Let everything air out for a day and I believe you will eliminate your odor. You may have to give everything an "odor" test before putting it back into your cabinets (even food boxes or cans but especially dishes).
You can now use some of the other suggestions about using something to mask the odor but I believe this will eliminate most of the source and make your area more enjoyable.