Well done
@kavil on getting what looks like a good car - so much less rust than cars here in UK & Ireland
Good news is, main bodies are galvanised so rust shouldn't be too much of a concern.
I agree with koalar - one of the most important rules of car maintenance is: if it isn't broken, don't fix it
I would only do what needs doing for now and leave everything else for later.
Old cars are typically overdue for engine oil and filter change - if the oil is black, do that.
Keep a very close eye on the oil level until you're happy it's not losing any.
Gearbox, check the oil level, if it needs topping up (to the level of the filler plug) may as well change it.
Gearbox oil spec seemed to change about a bit, so check your owners' handbook.
My 2004 1.2 was 75W-80 to API GL4 or GL5 (1.65l)
Filler plug 12mm hex (front of 'box) / drain plug 8mm hex (rear of 'box)
Check brake fluid - should be clear, if cloudy, change with DOT4 (also typically overdue).
Check coolant - should be pink or orange. If dirty, flush and replace.
Again, keep a close eye on coolant level until you are sure it is stable.
Temperature gauge should reach half way within a few minutes' driving. If it sits below half way, thermostat is leaking, which is common with age. Replace thermostat along with coolant if it needs it.
Check the 'duckbill' scuttle drains below the wipers are clear - they block easily.
Then, once a year I would check shock absorbers for leaks and check brakes:
Fronts: remove pads and clear rust off sliding surface of reaction frames:
https://www.fiatforum.com/threads/500-disc-brakes-problems-solutions.281466/
Rears: remove drums and clean, lift rubber boots of wheel cylinders to check for leaks. If dry, leave them alone
And keep on top of the areas that can rust:
- rear axle spring pans
- rear sills (access to treat rust via rubber bung in rear wheel arch)
- shock absorbers & springs
- mud trap behind front springs
- coolant return pipe across front of engine
- engine sump pan