You can open a basic bank account if you do not have the ID and proof of address that you need to open a current account. The Irish Banking Culture Board has a Guide to Basic Bank Accounts (pdf) in Ukrainian.
Many banks have made it easier to open a bank account if you have arrived in Ireland from Ukraine.
An Post (the Irish postal service) has current accounts. You need photo ID (you can use expired ID), your temporary permission letter and your PPS number. You need proof of address:
- A signed letter from the person you are staying with along with a household bill in their name
- A letter on headed paper if you are staying in temporary accommodation provided by the Irish state.