Wine glasses will be fine, even if you end up rushing out to wash them between courses. if that's going to be the case, keep a smooth surfaced towel handy for drying them, less lint to try to polish off of them.Please don't use the disposables, they affect the taste of the wine.
The port is traditionally served in either a large goblet or a snifter, which one seems to depend on location, but no one will be horrified at getting their port in a different glass. I try to avoid using jelly jars, but that's me

You don't say if your current glasses are the larger or smaller wine goblets?
When you do start collecting a wider array of stemware, buy flutes next,(sparkling wines lose their bubbles and their chill very quickly in those wide lipped flat ones, those are better for fancy-ing up puddings) then get the "other" winelass, then worry about snifters, sherry glasses,etc. If you buy closed stock, and care about matched stemware, get an extra pair for every 6 or 8 you think you'll need, and stash them away as replacement pieces.
One thing I used to get away with at my table was unmatched stems, I had a twenty or so glasses, in pairs or fours

...if you're serving,let's say, 8 people including yourself, you can set each dinner pairing different glasses, I alway started at my own setting and paired my glass with the gentleman at my right,with the lady to HIS right beginning a different pair and so on... it would be easier to show than explain, but you get the idea? If you have 6 of one type at a table of 8 diners, you and the fellow at your right would get the odd lot.
Best of luck with your party!