I have a bottle of unopened 23 year old corked spanish white cooking wine

Generally... not as quickly. The salt content of the wine is much higher than drinking wine, which will preserve it longer. It will still "go bad" eventually. (Going bad is more a decrease in quality. It will still be edible, but unpleasant. It won't make you sick, except by the bad taste!)

See this from a Wikipedia Article: "Cooking wine typically available in North America is treated with salt as a preservative and food coloring."

So, it's safer than normal wine, but I'd try and use it soon. The sooner, the better.

There is no trick, it just won't work. Synthetic corks are popular as replacements to cork not only because they are cheaper, but more effective at preserving wine as they don't dry out, and they expand more in the neck keeping a tighter seal. This makes it more difficult, if not impossible, to get them back in.

The simple and easy solution is to buy re-usable bottle stoppers. There are many different types, I prefer the ones which have a lever or button you push down to expand the stopper as they are best at preventing spills and leaks. There are vacuum sealers as well, however IMHO they're gimmicks and don't improve the storage of wine.