All that I know about Arizona Cypress, is "internet knowledge", I have never grown one. I know the Mendocino Cypress, a related Cupressus species, have been successfully used as bonsai. Look for photos in archives of Golden State Bonsai Newsletters. They turn up here and there. Photos look pretty good, superficially similar in general appearance to Shimpaku, usually without much deadwood in the design.
Your tree needs to grow and develop character. Your trunk and largest branches are arrow straight, linear. They need to be wired to put movement into them. BUT before you do so, you need to think about style, what is your model you would like to shape this tree toward? Other questions:
How tall do you want this tree to be?
How wide?
Typically the trunk an inch or so above the roots of a bonsai should be somewhere around 20% to 33 % of the greater, height or width. So a 12 inch tall tree should have a trunk around 2 inches to 4 inches in diameter to get the ''ancient'' tree image. Smaller diameter trunks give either a slender forest tree feel or a young tree feel. Your trunk needs to grow, so beyond putting some movement (bends or twists) into the trunk, you really just need to let it grow without pruning anything off. THe more leaves (scale needles) the more quickly the the trunk diameter increases.
You also need to train the roots. I would NEVER slip pot this tree. This tree needs to have a flat root system, that radiates horizontally, with roots coming off the trunk like spokes of a wheel. The "average" of the shelf bonsai pot is less than 4 inches deep. Usually between 2 & 3 inches deep. You need to develop a root system that could be fit into a bonsai pot without trauma when the time comes to move to a bonsai pot.
If you are unsure of what to do, you don't have to do anything about the trunk and branches just yet, you should start work on the roots. I would work to find an appropriate training pot, flat or box. I use Anderson flats, others use cut off nursery pots, some make boxes the right size. Anderson flats are 16 x 16 x 5 inches deep and I never put more than 4 inches of media in the flat, this holds about 3 gallons (nursery trade gallons). Do repot until you find an appropriate training flat. Also gather appropriate media. I would use the same sort of potting mix that is used for spruce. A pine mix can work. I would use a mix that is near half to three quarters pumice, the rest being either pine bark or Akadama. But you are in a different climate than I am, you may choose other media more appropriate to your area.
You might repot late summer, but at least 9 to 12 weeks before you first hard freezes, or you can wait an repot in spring. When you repot you comb out the roots and do your best to remove downward roots, if the is a tap root, remove that. Goal is to create a radial ''spokes of a wheel'' pattern that will fit in 3 inches of soil. I would also tilt the trunk so that it is not perfectly vertical coming out of the soil. Only formal upright styles will the trunk be perfectly vertical. Once repotted, no fertilizer for 3 to 6 weeks. Wait until you see some new growth. Then you can resume fertilizer. For a year after repotting, no pruning, no styling of the tree. Just let it get growing again.