Can the person to buy send me $1 bitcoin?
Yes. Though this would only be a good idea if you're a using a wallet whereas you can choose your own transaction fees.
Is Binance suitable for the process?
Binance[1] has a withdrawal fee of 0.0004 BTC, which is around $3.74 in current prices, so that really wouldn't be a good idea. In fact, withdrawing $1 on any exchange wouldn't be a good idea because exchanges has a constant withdrawal fee regardless of the withdrawal amount.
[1]
https://www.binance.com/en/fee/depositCan you make suggestion to bitcoin wallet? and thanks