Will I have to go through customs in Birmingham?
Yes you will have to go through UK customs* and immigration. The UK has a particular arrangement with the Republic of Ireland where flights to and from Ireland are treated almost as domestic flights by the British although not by the Irish. Therefore you clear UK immigration first, before going to Ireland where you will also clear Irish immigration. But before your flight to Ireland you will need to go through security at Birmingham too.
As such one hour between landing and the next take off is very tight. It's a small airport. If the stars aligned for you, it could be done. But I wouldn't deliberately book it.
If you are thinking of checking a bag, then this is becoming rapidly close to impossible.
I would suggest you book onto the later flight, and continue to have a back up plan if your transatlantic flight has a serious delay.
For this kind of thing I personally would allow about 150 minutes. I find that is usually long enough to absorb the typical delays that happen.
* Strictly speaking you don't have to go through UK customs if in international transit directly to Ireland, only immigration, but Birmingham is not set up for this special transfer, and in any case UK customs are a non-event.