Much depends on what engine it is. Some old engines will run for years on unleaded, some won't. It's generally accepted that old Triumph engines for example, although designed for leaded they are happy on unleaded, with no valve seat recession. Others are less happy and need that lead!
The poor running problem is another issue, more to do with the low octane rating of modern petrol: you need to reset the timing to cope with the low octane.
You can get round that by using one of the new fuels they brand 'ultimate' or similar....these are higher octane versions of normal unleaded. These fuels allow you to set your timing a bit closer to what the original settings were and the car will go a lot better, but you may still need to use a lead substitute.
I have a Triumph Vitesse, which is one of those cars designed for 5 star, and it runs a lot better on the modern superfuels with the timing back to the factory timing than it ever did on normal unleaded!!