If my EOS account run out of CPU, would it help if I run my custom node to process my own transactions?
If I'm required to become a block producer for this, what are the benefits of running hungry MainNet node?
No, it wouldn't help you. Only the block producer who processes your transaction will determine how much you are billed for your CPU, and this is always done on your account. it is one of the reasons EOS was built this way, to create buying pressure on the token.