Well, it's going to take a while.
We can assume we need about a minimum of 25,000 miles of cable (22,000 miles for geosync orbit, plus 3000 for the counterweight).
The elevator car will weigh about 100,000 lbs. Why? Because it's a round number in the range of the Space Shuttle's empty weight. The payload needs to support 40,000lbs, because that's the range of the first ISS module. So we need to support 140,000lbs of cargo.
For simplicity, let's just say we need 200,000 strands of hair 25,000 miles long, with 10% extra for joining and split ends. So 5.5 million miles of hair, or one 36,000 mile long head of superhair (see below).
Superman is superhuman, so while the average dark haired person has only about 110,000 hairs on his head, he has 150,000. Each hair grows at 0.04cm/day or about 0.016in/day. That's $2.5 \times 10^{-7}$ miles per day. We'll need his whole head for the weight we need to support.
Google says that's 144,000,000,000 days. That's just shy of 400,000,000 years. By that time, humans could be a type III civilization, rendering the whole thing moot.
Just for the sake of an argument, let's say superman's hair is infinitely strong, like the magic alien spear. Now a single strand can support the whole weight of the elevator system.
We still need 36,000 miles, but now the hair grows at 0.037 miles/day! And since hairgrowing is one of his superpowers, he can do it 500× faster than the average human. So now we have 1.86 miles every day! In that case it will take us only 53 years to grow a space elevator. Since superman has been around since the 1930s, we should have enough of his hair laying around to make this work!
Just be glad superman isn't a redhead.