A good question and impossible to give a definitive answer, but here goes anyway. This is only toddys opinion.
Neo-Geo carts are very reliable from my experience. You are more likely to experience problems with the mobo hardware.
If the carts are treated well, i.e. keep them free of dust and any other foreign objects/substances and do not expose to extremes of temp (keep the system ventilated, but dont get anal about it, in reality they should work just as well at the Poles as in the Sahara) or UV light, do not let them sit in direct sunlight.
If there are eproms in the cart they can be partially erased with exposure, having said this the case will shield them unless it is transparent as with some MVS carts,
they will probably outlive you.
On the UV issue, if there are eproms on the cart boards, it is not advisable to photograph them with camera flash on, unless the windows are covered up with tape. It is unlikely that there will be partial erasure, but why risk it.
Also excessive supply voltage may cause problems (perhaps as a result of a faulty psu especially with mvs), as with the mother board.
Static electricity is a potential problem, be aware of those issues.
There is also a phenomenon (some might argue real or theoretical) known as "bit rot" which is "facetiously described" here:
http://en.wikipedia.org/wiki/Bit_rot 
So in theory, in perhaps tens, hundreds or thousands of years time the roms/eproms data may become corrupt as a result?
As is the nature with all electronics, there may be some unexpected component failure (say a rom chip) which is beyond the designers, manufacturers or users control. The chances are remote; it’s usually down to bad luck
but it does happen.
As far as storage is concerned, ideally a cool dark dry place.
Bottom line, the cart will probably out-live the user. I would guess by hundreds of years. But please don’t quote me
Edit: As a foot note, my only cart problems have been climatic related.
Living on the coast in the tropics I have come across a couple of carts (out of about 40 total over a 4 year period) which had open traces near the cart edge connectors as a result of corrosion of tracks, imho caused by a combination of high humidity and salty air.
Never buy any carts from toddy.
