Buy a banger for £200 with a years MOT.
This. I've never ever paid more than £1,000 for my cars (including a year's insurance) and I've managed just fine. I drive a 58 reg now, but I really enjoyed my time in 'bangers' - so much so I'd seriously consider doing it again!
So long as you know the big things to watch out for, or have a savvy mate, you're well in. Looks mean nothing, and there's not really such thing as a 'cool' banger (depending on your definition of cool though, I suppose). As was said above, we're talking tools for a job here.
Things like low-ish mileage Pug 405s with a 1.9td lump are worth looking at as they're only popular in their 'niche' (read: cheap) and cheap to insure. They pull well, do 50mpg on a run and have a nice turbo. Check electrics before purchase, though I mean windows and such. Don't ever expect to find working RCL in them as that invariably dies long before you buy it. Also wishbones, as they're a £55 a side fix even if you fit yourself (compared to £17 for an Astra part for example). Also make sure the coolant system works OK and the fan doesn't come on immediately on start-up. Fifth gear can bounce on them, so make sure there's no movement when you apply/remove throttle in fifth.
That sounds a lot but it's nothing really major. Just make sure the head's decent, it still pulls well and doesn't get over 90oC (but make sure the temp gauge DOES go up from cold!) and you should have a good one. A decent one can be had for £300 to £400 easily on a L to N reg with <175k on the clock and will easily last a year. I did 55k in a year in my old 405 estate and it still went like new. The guy I sold it to (with 265,000 on the clock) drove it to Russia and back (honestly!) that year and it's still going to this day.
1.4 three door Astra petrols are cheap to run and insure, and are "chuck around, scrape the sides, who really cares?" cheap. I snapped a cam belt on mine (car bought for £150) and the scrap yard delivered and fitted a 'new' engine for £100 that lasted me another 20k before I sold it at a profit. Or a nice 5 door (maybe estate if you can find a decent one) with a bigger engine or a derv lump.
Ditto mid-90s Mondeo diesels and, if you can find one, a nice 1.7td (Isuzu lump) Cavalier or early Vectra. Yeah they're Vauxhalls again, but they're bloody cheap and run on 100% veggie oil straight out of Tesco (as do 405s with a Bosch IP btw) at significant savings. All legal, too.
My brother bought a H reg Honda Concerto 1.6 VTEC with full leather and 60k on the clock - literally unblemished in every way and FSH. Price? £450. Primeras are worth a look, as are old Passats (diesel). There's plenty of motors that can run well for no money for the sake of 12 months.
Basically, if you CAN'T buy and run a car for a year on a £1,000 budget you're doing it wrong. Sure, things occasionally go wrong and things explode

But so long as you have breakdown cover and buy sensibly to begin with you should never have to use it. I called RAC out once in three years (>150,000 miles) of bargain motoring - and only then because I had a wishbone snap due to a bloody big, unavoidable pothole on a DC.
This type of motoring will probably generate strange noises from the BMW/anti VX/high-end set, but for what you're after the above holds true. Just don't make the mistake I did early on... No expensive services (oil changes etc at home are <£20 and worth it though), and if something goes bang and will cost more than £50 to fix, scrap it and BUY A "NEW" CAR.

Personally, I'd buy a "British" car like a Mondeo or Astra. There's more to choose from, you can afford to walk away from 10 bad ones to find 1 exceptional one, and parts and labour are silly cheap at this end of the market should something minor go wrong.