It's great that you are coming to Dublin this August with your friend!
Since renting a car
seems tricky in your case, I would suggest using public transport
to go around the island.
While 5 days seems a bit short to see all of the island, you could for instance base yourself in Dublin and go on day trips
(using public transport or tour operators) or build an itinerary across different locations.
I suggest you use the Transport for Ireland
website and the Rome2Rio
website to plan your trip using public transport. Furthermore, here are trip ideas
and suggestions for a 3 days trip
right outside Dublin.
Here's also a search tool
for accommodation you might find useful. You can search for hostels and B&B's if you decide to stay in different locations.
Were there some locations you absolutely wanted to see during your time with us?