Should a Honda Elite 80 Scooter still be able to run after sitting idle for 5 years?

What kind of maintenance should be done besides a new battery and spark plug?

What kind of problems can I expect?

Take out the spark plug and put a squirt of engine oil in the
hole.
Check the oil level.
leave the plug out and rotate the engine SLOWLY by hand a few times
insert the plug
put fresh fuel in it (dont use the old stuff)
Start it up (yea it will smoke)
check for fuel leaks and oil leaks.

New tires-5 years of sitting means probable dry rot. Even if they look brand new. You don't need the tires to blow while in motion.

Carb job-if the tank wasn't properly drained, or the bike wasn't prepared for storage then the gas may have turned into 'varnish'.

Oil and filter change.

As someone else said, seals and rubbers may be at risk to fail, so have a good tuneup done. The mechanic needs to know the bike has sat 5 years so a sharp eye can be kept for any other issues.

Hell yes, I found one a few years back on the curb & the sticker said that it was mfg in 1975. I drilled a small hole in the air filter housing on the side where your right foot goes greased everything up and shot some starter fluid & kicked it, it started right up & has been so much fun ever since, I can even get my wife on it & she is such a prude about everything. Have fun & good luck. You may have to shoot fluid if you don' ride it frequently.

Change the oil and clean out the gas tank before you refill it.

Check out all the other lubrications, grease for the bearings and gear oil in the transmission.

Get an owner's manual and a repair shop instruction book and read them just to be sure there isn't any speical needs for that model.

check for rusted gas tank. you can get those boiled out at a radiator shop. check your gaskets, they may be dried and cracked. anything rubber must be suspect (again drying and cracking). replace oil and lubricate everything.

have fun.

did you drain the gas out ?