To be fully stable, it will take several weeks to cycle. If you choose to cycle without a living creature in it, it will likely do a spike when you actually put your Axolotl in anyway.
You can buy a couple feed fish to cycle your tank, but this comes with it's own challenges. This is what I chose to do, and I ended up buying a new tank for my axolotl. Why? First of all, my feeder fish had a fin disease. I treated it and it seems to have disappeared, but in the back of my head I'm thinking: "What if they just built up immunities? What if they are still carriers and will give it to my axolotl?" And I've had my goldfish for eight months. Time to identify and treat diseases. You, with two months, might not have that opportunity. And that brings up another thing. What happens to the goldfish after you're done cycling? Are you going to keep it? Where? It's not a good idea to let your Axolotl eat it. I now have a 29 gallon tank for 7 goldfish, who are rapidly outgrowing it...
You can just let the water sit for a few days to a week (let all the chlorine evaporate) and then cycle your tank with your axolotl. But you'll have to watch the water parameters very carefully. This would however, be the quickest option. it would take a week or two.
Personally, I've decided to cycle my (new) tank with shrimp. Not sure if it'll work completely, but I figure it will lesson the impact when I put my axolotl in, and my axolotl can eat the shrimp safely. This way will take longer. A few weeks for the shrimp, and then put my axolotl in and it'll take another two weeks or so.
