Minor recommended changes to your map, mostly just making sure you're hugging the southern coastline. Those peninsulas are wild, rugged, and beautiful -- you'll want to bike as close to them as possible!
Now, a word of caution. I only mention this as I'm unsure of how much you've looked into this, so this may come as no surprise: The vast majority of roads in Ireland are very narrow. Most don't have shoulders. The roads are narrow and where they end will often be an immediate drop-off, ancient stoned fence (there's beautiful stone fences everywhere, covered in ivy). So there isn't a lot of room for bikers. On the plus side, many roads are sparsely traveled. But I imagine it's something you should be aware of.
Also, I'd say skip the east coast. It's less stunning than the west coast. If anything, add Connemara/Co. Mayo (north and north west of Galway (adjusted map to show recommended additions).
Buying a used bike will probably be your best bet. I've not done a biking tour so I can't recommend any specifics like that. Do you have a new iPhone or Android that's unlocked? If so, be sure to get a local SIM card so you can email or call B&B hosts/hotels to let them know if you'll be late due to biking delays (B&B hosts in particular are picky about arrival times).
Some ideas:
You could take a boat tour to see the Cliffs of Moher and then get dropped over on Inishmore to bike around the Aran Islands. That'd be pretty unique/awesome. Then take a boat north and get dropped off in Galway (so it stays in line with your general route).
Not sure how you feel about hostels, but if you're down be sure to check out Black Valley Hostel. It's on the valley floor of Killarney National Park, surrounded by stunning scenery. You could easily spend days there hiking and soaking in the beauty of the place. Although, I will note that the roads are fairly windy and the main road (not the one the hostel is on) through Killarney National Park is fairly popular so you WILL run into other vehicles, maybe even a tourist bus. And it's narrow with no shoulders.
Not far from the hostel is Kate Kearney's Cottage, which is an awesome drive/ride from the hostel and has great food. It's relatively secluded, the patrons are generally hikers/locals or the occasional tourist group that knows of it. It's out of the way for more 'touristy' hotspots, but the food is awesome and the people incredibly nice.
The weather! I should've mentioned this earlier. Ireland's weather is fickle, in a word. It will be raining on you, but you can see clear, gorgeous skies in the distance. Then it will be sunny. Then raining again. It changes through the day, so you'll need to be prepared to put on/take of rain coat, etc. Also, the rain is sometimes sideways due to the coastal winds -- so you might just anticipate getting rained on, regardless of prevention efforts!
That's all I can think of at the moment. Let me know if you have any specific questions!
Cheers.