You'll get a shower at their tiny Aer Lingus arrivals lounge beside the baggage carousel. If your onward back to the UK is with EI, you should also be able to use the shower in the main EI lounge while you wait for your flight (I think the showers in that lounge have reopened).

Better than BA lounge is subjective. EI's lounges are much smaller and have far less (very little!) food, but are much quieter, cleaner and quicker to navigate. I love EI's lounge because I fly short haul every week, and they make a perfect touchdown zone for 20 mins before my flight boards. For this reason, I'd take them over BA's any day of the week, but like with the J seat, it's probably a matter of personal preference and use case.