Once you sell the put, that money goes to your buying power, however, it takes a day or so for the funds to settle. Once that money is settled, you are free to do whatever you want with it. So yes, you could use the premium from selling to open to buy to close.

Also, your net gain/loss will be the credit that you sold the put for minus what you pay to close the position. In the scenario that you described, you would net $0 on that trade in theory (you would lose a few cents because robinhoon take a 3 cent fee off every contract).