Don’t need a pet passport for taking pets between the UK and here. The rabies vaccination would be needed for travel outside UK and Ireland. There is an initial rabies shot and a very tight window for getting the booster at a vets here. If you miss out on the booster you have to begin from scratch. It is a costly process, your vet may not issue you with a reminder.
There may be local restrictions in the country you visit mainland Europe but I’ve no knowledge on that part. Try irishanimals.com.