you can buy them online using your creditdebit card on http://prizebonds.ie/ or at any post-office.I think at some point they will ask you to prove your identity (passport, prove of address, etc)
edit, according to their site, the can only accept debit cards on their site. I think its the same at Post Office counters - they will only take debit cards as payments for anything, not credit cards.
(a) Monthly Direct Debit form available online at www.StateSavings.ie
(b) Online www.StateSavings.ie
(c) Telephone State Savings ™ 1850 30 50 60.
Our agent will record your details. Payment by Debit Card (Visa Debit, Maestro or Debit MasterCard).
(d) Freepost Application Form
Print an application form from www.StateSavings.ie and complete it and mail it to
State Savings, GPO, FREEPOST, Dublin 1 (no stamp required)
or deliver it to
(e) Any Post Office
I'd check out some of the very good posts on AAM regarding investing in Prizebonds before committing to it - food for thought!