A 4 litre XJ-S will have the AJ6 engine also found in the XJ40. This is virtually indestructible - it's one of the most durable engines fitted to anything and will shrug off a lack of servicing and abuse in a way the V12 will not.
Incidentally, the AJ6 engine was subject to an in-depth article in Car magazine where it was described very favourably by LJK Setright as part of his technical assessment of the Jaguar XJ40 - a car Setright consistently admired, long after it had become unfashionable elsewhere. He was less kind about the Series Jaguars though.
There isn't much that goes wrong with AJ6 - the engine nearly always will outlast the body of the car. Smoke at start up is a common fault - the valve guide seals leak and most will smoke slightly when started after a lay up now. It should clear quickly - really just a puff - and then not smoke at other times when you drive.
Oil pressure problems are virtually unknown on the AJ6 - the oil pressure sender is however quite a flaky little thing and can fail at will, reading low or bouncing about. Low oil pressure on the gauge is almost certainly the sender.
The automatic transmission is a ZF 4HP24 - not the last word in sophistication, but remarkably strong and trouble free in these Jags. Any jerkiness will nearly always be cured with a fluid and filter change.
The manual XJ-S used a Getrag 290. The clutch is slightly heavy on them as is the gear change - but they perform well and give little trouble. They are much less common than the automatic.
The face lifted XJS (face lifted cars lost the hyphen) will have the AJ16 engine found in the Jaguar X300. This is a smoother engine than the AJ6 (which is itself a decently smooth thing) and slightly more powerful. It is basically the same engine though. Transmissions are identical. The significant mechanical change on face lifted cars is the switch to outboard rear disc brakes - rather than Series Jaguar-style inboard disc brakes ion the earlier cars. These are much easier to maintain.
The face lifted cars are less rust prone than the earlier cars (which can be shockingly rusty) - and generally regarded as better engineered. Whilst I recognise they are better, I don't like them as much as the earlier 4 litre. They are less attractive and seem to have lost some of the earlier car's swagger. Face lifted cars are capable of suffering serious rust - the problem was not solved in them - though they are less affected than the pre-facelift.
Any XJ-S rusts pretty badly - and they seem to have spaghetti electrics too. Most of my limited knowledge of the XJ-S is from reading owner's postings on them on the XJ40 site I am a member of (I am a long term '40 owner). There is quite a bit of love for the big Jag coupe there as you would expect - but also a consensus that the XJ-S is considerably more likely to rust than an XJ40 and its electrics are nothing like so reliable as the XJ40 either. Given this, check all the electircs work - and that the carpets are dry. Wet means rust!
I do love them though.