Possibly a Trichocereus. If it was mine I'd leave it in the position it is in in the picture and maybe remove the pup with a sharp blade. This can be left to callus over then try potting it on in dry sand till it roots. It will not need watering to grow new roots. The main plant will grow more pups in due time. If you get viable new plants the old one could be ditched if you don't feel too sentimental about it.
The above propagation method works with most cacti.

Small story... I had a cactus and it started to get dry just right above the soil surface, this area did not increase in size. The "neck" became too thin to keep whole cactus body, so one windy day that place broke off and cactus fell down. I thought it is done, but for some reasons I did not get rid of it, the cactus kept laying for few weeks and one day I've noticed the wound has dried and small roots started to appear.
What I'm going to tell is to cut the top straightest part of your cactus and let it grow new roots. Would be great to get advice from an expert as for this trick.

P.S. My cactus is doing well since that case, blooms every year... it was 3 years ago or so.

This post was edited by ibilous on Sat, Aug 2, 14 at 6:34

In addition to the advice already given, you could visit the GW
Cacti & Succulents Forum.

Here is a link that might be useful: Cacti & Succulents Forum

I'm no Cacti expert, but I grow a few.
I think it fell over because you're watering it too much.
I can tell by the thriving Oxalis in the same pot.
Even the Jade is getting too much water.
Mike