Hi VUK, welcome to the forum!
Yeah, those balljoints are a common failure. I've just bought two new ones, even though I found out today that both mine were actually OK.
A simple-to-rectify fault is the rear lights. Not usually a blown bulb, often a case of unplugging the plug and cleaning it, and tweaking all the bulb holders. The most often-blown bulbs are the 5W jobbies (number plate, side repeaters (remove from inside the wing to avoid snapping the light off) and front parking lights).
Brakes, I find, are a common problem - specifically the balance of the back brakes and the handbrake.
Rust - of course - particularly under the windscreen, at the edges of the floor, and in the door frame near the hinges. Under the front wing tops, about a third of the way forward from the windscreen, there is a rust trap near the suspension turret. Inside the back panel, hidden by the rear bumper (though this is hardly likely to be an MOT fail point).
As these cars get older, it's beginning to be more a case of where they *don't* rust than where they do. When I think about it, I have seen (on separate Unos) rust pretty much everywhere, except the bulkhead. It is also not a case of an Uno that has rust in one place being likely to have rust elsewhere. I've seen cases where one door was practically missing the bottom, the other three doors were perfect. This makes it different to 1970's cars!
On a long trip (to return home) you shouldn't take chances with tyres. Check them very carefully and if applicable try to leave enough time to get the tyres replaced before you drive home. In my experience, Unos for sale typically sit around for ages beforehand, the tyres steadily perishing or developing carbuncles.
The worst feature of the Uno that I bought last year was the cut springs in the front suspension - one had already slipped off so I fastened them on with cable ties - the ultra-low suspension had broken the tripode driveshaft joint circlips and all the needle rollers had been lost in the transmission, giving an awful wobbling/vibration. Also the gearbox had a cracked mounting from accident damage.
Driving it home (about 130km) was hair-raising - when moving off there would be a loud BANG and a feeling of the front end jumping off the road (I later realised this was the driveshaft jumping out of, and back into, the diff.) This was after bypassing a completely split junction at the heater hose that caused most of the coolant to pour out. It's just amazing how bad cars can get with total neglect and thoughtless abuse.
Basically, I guess I am suggesting to take a reasonable assortment of tools (a 12mm deep socket served as a hose joiner!), cable ties and duct tape, and a mobile phone... but you will get it home somehow. Good luck with the MOT!
-Alex