Correct me if I'm wrong - are you talking about someone you'd like to become closer to over the next year? If so, I'd probably say something like:
I hope we get to know each other better during that time.
I hope we become closer during that time.
When you talk about "getting closer", that often has a romantic undertone, so it's more appropriate with someone you have a romantic interest in. Friends like to "get to know" each other and "stay in touch" so that they are in each other's lives.
In terms of usage, when you express hope about something in the future, you don't need to explicitly use the future tense, since the verb "hope" itself implies a future event in this case:
- I hope I see you at the bar tonight.
- I hope she gets the job she really wants.
- I hope your team wins the championship next week.
Expressions with "get to", like in your example "get to be closer", usually mean "get a chance to":
- "I hope I get to see you before you leave" means "I hope I have the chance to see you before you leave."
- "I'll never get to drive such an expensive car" similarly means "I'll never have the chance to drive such an expensive car."
So "I hope we get to be close" sounds like "I hope we have a chance to be close", but I think what you mean is that you want to become close. "I hope we are close" is the kind of thing you would tell the parents of someone you are marrying, or the son/daughter you didn't know you had and just met, and in that context it means "I hope we're not going to be strangers, but rather that we're going to treat each other as people who are close".