It looks good to me. I would have no clue what name to give it except for cactus. There is no rush to uproot this plant, it is still quite young and looks like it has plenty of space (maybe share a picture of the container from further back so we can see the dimensions).
If this is an indoor plant, it will enjoy as much natural light as you can provide. Right in front of a sunny southerly facing window would probably be ideal. Hours of daily sun year round, weather permitting, with no curtain or blinds in the way. I'm not sure where it's been, but that's a good guideline for most indoor cacti once they are old enough to resemble an actual cactus. As for water, it would be good to saturate the soil (assuming the water has an exit through holes at the bottom) and then allow it to dry out halfway or more at depth, but not sit bone dry for any extended period. Baby cacti tend to be more tolerant of extra water and less tolerant of bone dry soil, compared to their adult counterparts, but those are general guidelines.
With a picture of the container, some info about the exposure, and some info about how much and how often you have been watering, we might be able to give more specific advice.