but I don't understand the dollar amount they show with it and how it relates to my specific situation (i.e. right now it shows medium priority 4 sat/vB is $0.58).
The fiat values of fees displayed on mempool.space are for a normal segwit transaction with 1 input and 2 output which is around 140 vbytes in size.
With the fee rate of 4 sat/vbyte for such transaction, the absolute fee would be 4 × 140 = 560 sat (around $0.58 at the current rates)
Your transaction will include 1 legacy input and 1 segwit output and will be around 190 vbyte.
So, if you set the fee rate to 4 sat/vbyte, the absolute fee would be around 4 × 190 = 760 sat (around $0.79 at the current rates).
For example, using your example above, if it was 4 sat/vB and I paid 760 sat as a transaction fee, how much would be deducted from 0.3 BTC?
760 sat equals to 0.0000076 BTC and that will be all the fee you will pay for your transaction.
Again, the transaction fee doesn't depend on the amount of bitcoin you send.
The transaction fee depends on the fee rate and the (virtual) size of you transaction and it doesn't matter whether you send 0.001 BTC, 0.3 BTC or 10 BTC.