I need a little more help. So I talked to the lady at the entrance and she gave me a weak spell to use and get inside... however she is mean and gave me an apprentice spell which I have nowhere near enough magicka to cast.

Anything I can do?

Try reloading a save and pick different dialogue options. I've heard sometimes you only need use a shout to get in.

Get a extra magica potion

Level your magicka first...

This. You can even take the same dialogue options. The spell she requests you to cast is picked randomly from a list. If you don't have it you can buy it from her for 30 gold. Rinse and repeat until you get a spell you can cast (and which is useful to you...).

Fortify magic potion anyone?

If you want to get into the college, and can't even cast one apprentice spell, it is not the place for you

Another option is to go through the main quest until you reach the part you need to enter the guild and you will have an option to prove you are the dragonborn, but that might not appeal to you so try reloading the save.

Find an alchemist and buy a potion to temperarily increase your magicka.

just wait until you raise enough magicka through leveling. Or an amulet to raise magicka, either way it doesn't take too many levels, of course I always at least find a use for magicka if your say a warrior build, resto for one.