It is definitely possible to reach level 2 in potions by that time (bearing in mind your potions level rises when you craft a potion as well as when you study) but the deadline is admittedly very tight! In the full game I'll have Pandora make her request earlier to give the player more time to raise their potion skill, I've had other feedback to the same effect so you're definitely not the only one having trouble.
As far as weapons go, you'll learn some by raising your metalworking stat, but the easiest way to get early game weapons is to either fulfil Ingrid's request for cloth (she gives you a weapons magazine as a reward, admittedly I think it would be difficult to craft the cloth she requests before the demo ends) or to fulfil the request that gives you Water Balloons as a reward. You'll be able to use those as a weapon in the demo, but you can also see the recipe for any item once you have it in your inventory, and you eventually unlock the ability to craft items freely even if you haven't learned the recipe in class (I think you might be able to unlock this in the demo if you fulfil a lot of requests, but I'm not actually sure...! It unlocks automatically a little after the demo ends). I'm currently working on adding more weapons to the game, so there should be lots to play around with in the full release.
Phew, sorry for the essay! 😅