It strongly hinges on your definition of "fossilised" as delivered in dialog by a character who does not know what he is talking about.
As the word can be associated with many varieties of preservation, it then comes down to applying the definitions that fit the best. Looking at these definitions, permineralisation, crystalisation, casting and other commonly understood methods of fossilisation require a lot of time. Let's say, 10,000 years at a minimum. Typically longer.
On the other hand, there's one other EXTREMELY significant factor that rules out such commonly appreciated means of preservation that may apply here. The sample must be buried for ANY of these processes to occur.
As the Space Jockey has NEVER been buried, everyone of these processes are instantly disqualified and eliminated.
That still leaves a number of other preservation possibilities. We can rule out bioimmuration as well for similar reasons. The Space Jockey has never been coated by another organism for this process to occur.
What remains are two very likely candidates of preservation: Desiccation and freezing. Desiccation is the drying out of a sample and freezing is, well, if you don't understand what freezing is you're at least a century behind the rest of us. lol.
Both of these are likely candidates and both can be applied to the Space Jockey; and both can be considered as "fossilised" in terms of preservation. After crashing, the dry interior of the derelict ship is an ideal place to preserve the pilot as it dries out immediately after the crash and become desiccated. Cold can also serve as a factor here.
In short, desiccation is very likely to be the cause of the pilot's preservation. It's relatively quick, too. Large animals dying in hot environments take about a week to fully dry out and take on the appearance of being dead for a lot longer than they really are.
So, what exactly is the age of the Space Jockey? It could easily be any age you care to nominate. Given the evidence provided it could easily be anything from 10,000,000yo to 10yo.
Possibly even less.