Credit score help please…?

I am going to try and buy a house in the near future. My credit score is at the bottom of the required number to get a mortgage loan. I have very little “new” credit, some student loans that my first payment is due on later this month, one credit card (zero late payments, but is maxed out, low $ 500 limit). How fast will my score come up (how many pts in how much time) if I pay down my balance and make all my student loan payments on time? Also, how many points would a credit score drop for a “hard hit”? My score is in the 620’s.

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