Replace peppermint extract for peppermint syrup?

Yes, you can use the syrup or you can make your own extracts!

to use the syrup, experiment by using a teaspoon of the syrup then if the flavor isn't strong enough, add a bit more until you have the taste you wish.

to make extract, get a good quality brandy, vodka, etc and stuff the mint leaves in a bottle you can seal then fill with the alcohol. Put in dark corner for around 2 to 3 months to let it absorb the flavor of the herb. Use as called for in recipes. This works for any herb!! Break vanilla beans up and put in bottle for vanilla extract.

Extract is the pure flavor, usually in a really concentrated in alcohol. It is really quite strong. Syrup implies that the flavor is diluted in sugar. So depending on the recipe, you may be able to fix it. You would have to adjust the amount of sugar you add (because it is in the syrup) and accept that the mint flavor will not be nearly as strong. But many times the syrup will not be a very good substitution.

If you can't find peppermint extract, I would say your best substitution is some other extract...vanilla, or spearmint or lemon, or orange, or whatever you can find, or whatever seems most acceptable in your recipe.

You can make your own flavor extracts, if you can get your hands on actual peppermint leaves, or if you can find a way to grow it yourself...do so. Gather a bunch of leaves together, rough chop them and then cover with vodka, and set it aside for a couple months in a dark cabinet. The peppermint will infuse into the vodka. Filter off the leaves and discard them. The vodka will be strongly flavored of peppermint and you can use it by the teaspoon in your recipes.

You can do the same with vanilla beans, or lemon or orange peels....and you can also infuse things like garlic and basil into olive oil for your savory cooking. It is all the same method....flavor agent...carrier (what you infuse into)...time.

Peppermint Syrup

That would depend on what your preparing. If it is some type of pastry likes cookies, cakes, pies etc, I would say no. Peppermint extract is much stronger and condensed than peppermint syrup. You would have to add quite a bit more peppermint syrup than extract, which would make your batter too liquid, same goes for some sauces and savory dishes etc. If you are making some type of drink, you can probably adjust the ingredients to use the peppermint syrup. For a dessert sauce you can also adjust the ingredients and thicken the sauce by making a slurry from some of the syrup and corn starch.