It is "ar imeall chathair na Gaillimhe".
It would be wrong to write "cathair" or "chathrach" or "cathrach" here.
When you have 3 words in the same group like here, you don't always need to put 2 of the in the genitive case.
imeall + cathair na Gaillimhe > in this case you can't put cathair in the genitive so you lenite it instead.
If you say "ar imeall cathrach na Gaillimhe", it means "on the town-edge of Galway", so you link "imeall cathrach" + "Gaillimh". (a town-edge & Galway).
By the way, "tuiseal ginideach" would be pronounced more like "tishle guin-yidd-yah".
Remember that ch is NEVER pronounced as k in Irish (I mean, when you pronounce it properly! Most Irish people don't pronounce Irish properly because it isn't well taught).
Tchífidh mé sibh