Immigration in DUB on Way to UK?

Assuming you have this as one through ticket, the computer wouldn't have allowed you to book it unless there was enough time.
And if it is one through ticket, your luggage will be labelled through to Birmingham, you won't see it in Dublin.
In Birmingham, you will go through immigration (passports), pick up your luggage and go through customs in that order.
If you are lucky you will stay "airside" in Dublin and not go through Irish immigration.

Not sure where you want to go after Birmingham. From the terminal there is a short "people mover" shuttle to the railway station, from which you can catch trains into the city, to London, or to other locations.

I think you go through immigration in Dublin, which is where you enter the Common Travel Area, catch your connecting flight to Birmingham, do NOT go though UK passport control, but simply collect your checked luggage.

You haven't stated the time of your flights, but I would expect it would be at a time when you will be able to use the transfer corridor immigration which usually is quite quick. Do not go landside unless this is closed. If there isn't enough time (for example if the inward flight to DUB is late) then it will be down to Aer Lingus to sort it out if you are on a single ticket. You won't go through an immigration step on arrival in Birmingham.

Thanks for the replies. I am on one ticket and arrive early afternoon. Your info indicates I should be OK. I was thinking if I got held up there, it could result in a trickle down problem...cars, hotels, drinking real ale.