Never despair,
Vintage Port is fairly robust, being that it is fortified with brandy.
Unless it has been 'stood up' for years and the cork totally dried out, that is.
I would suggest completely uncovering the cork top (this will be messy with debris) and they usually have more than one layer of capsule as well.
If you can, before using the corkscrew, carefully (even use leather gloves) run a slim flexible sharp knife (I used to use a fish filleting knife) around between the cork top and the bottle neck, this is where they get most stuck.
In the end, if the cork goes into the bottle and Port, just pour/filter it all through a coffee filter paper, or even twin-ply kitchen roll.
'87 was a 'ripe' year so it may already naturally have gone tawny anyway, so don't be put off by the colour not still being bright red.
Best of luck and hope it's enjoyable.