My wife and I live in a 2-bedroom apartment. We think that a minicrib is a more is a more space efficient option than a full size crib but we are concerned that our child will outgrow it too fast. Does anyone have experience with smaller cribs and how old their child was when they left?
Assuming you mean an actual minicrib, which in the US has a size of aroiund 38 inches long by 23 inches wide (for the mattress), this will depend on your child heavily. (For comparison, a full/standard sized crib is 58 inches by 28 inches.) (Note: minicribs seem to vary some, particularly on the width. Make sure the crib you buy and the mattress you buy are intended for each other size-wise - there should be no space around the mattress.)
The average American child is a bit under 3 feet tall (36") at 2 years old. As such, unless your family tends towards the tall end of things, he will be able to lie lengthwise on the bed until he's about 2.
However, kids tend to be movers, and that's where this gets complicated. Some children will only use a corner of whatever crib you give them - and so wouldn't notice the difference. Some children only get a good night's sleep when they can roll about the crib/bed extensively, and need that extra space, in particular the width (a minicrib is around 15% narrower than a full crib).
Minicribs aren't likely to be substantially less safe than a full crib, as opposed to a pack-and-play or a bassinet, which are less safe at the point when the baby can move around some (both due to an easier time getting out/collapsing, and for SIDS reasons). Since they're built like a regular crib, they should be as safe (as long as you get the right size of crib mattress). I couldn't find any evidence of safety or lack thereof specific to minicribs, though, so this is just my reasoning.
Some research of comment threads online, like this blog post, suggest that mini-cribs work for some until they graduate to a toddler bed, while others do not. That's the real goal here, of course: make it to the toddler bed, at which point you dump the crib and get a toddler bed. You will need a separate toddler bed, likely, unless you manage to find a convertible that converts to using the larger mattress (which does exist). You may have to convert early (though we converted both of our children by 18 months with no issues).