Yes, move it now. Warm weather is the best time to repot all cacti.
Use your native soil from your garden in the pot. Soil is better than "cactus mixes" and other kinds of mixes containing a lot of organic matter. These break down in a couple of years; you must repot then or the plant suffers. In real soil you can keep your plant in the same pot for many years. I use coffee filters, whole or torn into pieces, at the bottoms of my pots, to keep soil in.
Replant a little deeper if you like, but not a lot deeper. It would be OK to lean it against the pot rim; they often grow like this on their own. Pups come from the side facing up.
A nice, large plant like that could easily go into a 1-gallon / 3.78 liter nursery container. I would use nothing less than a 4" / 10cm pot. They grow best if the soil stays moist all summer in our climate, and smaller pots dry out too fast. I don't use clay for any cacti; they dry out too fast here. In a wet climate like Florida I would use clay for all my cacti.
It will grow and bloom without fertilizer if it's in soil, but grows and blooms better with plentiful fertilizing. The more you fertilize, the more flowers you get. In native soil you can use ammonium sulfate at 1 tablespoon per gallon at every watering, if you have time. Water once a month with plain water to rinse out salts. Plants in nutrient-poor bagged "cactus mix" must be fertilized regularly or they hardly grow and rarely bloom.
These can be accustomed to full Phoenix sun if well watered, but it is safer to grow them on the east side of something or with afternoon shade.
If you bring the plant into the house on the night the flowers open, they will stay open for 2-3 days.
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