I have no idea what situation you are in, but showing up just for 30 minutes is a "fuck that shit" thing to me. It sounds like your boss is trying to micromanage shit to the max, and this is beyond "gonna get me some popcorn" moment of "efficiency". At that point, it's just a matter of time before shit hits the fan, spectacularly might I add.
If you guys need to share lunch between the two of you, you need quite a bit of time overlapping your work hours. And I don't mean that in the "you take 30 minutes off, we expect you back in 30 minutes, or else everything will fail critically" kind of way. The boss setting this up is absolutely retarded if this is the way they do things. One emergency, one accident, one anything, and you can kiss business goodbye (and many times in spectacular fashions).
As for my advice, no fucking clue. Usually I get good with the other workers, and we make things work. But as I've said, this is a catastrophe waiting to happen. Best case for you is everything goes as planned, worst case is company goes under while you're the only one on the clock. Well, maybe not that bad, but trust me, people will make it seem that way.
The word "misogyny" is like a bad joke, dropped at the first sign of trouble.