There is not set time for cycling. It depends on the tank. Some will take 2 weeks, others can take a month or even more. You'll need to get a testing kit, if you don't have one yet, so you can monitor the cycle. Also, if you're doing fishless cycling (which I strongly recommend) you'll need some pure ammonia too.
If you want to put fish in during cycling (which I would not recommend, as it is very stressful for them and could potentially kill them), you'll need to keep a close eye on the parameters and do water changes frequently to keep the levels tolerable for the fish.
Adding water from a mature tank won't do all that much. Not much bacteria is actually in the water. If you really want to kickstart it, use the filter from your mature tank. Or, put the filter from the new tank and run it alongside the one in the mature tank for a week or so, and you should be pretty much set. Just continue to keep an eye on the parameters and do water changes as needed.
Are the parameters in the other tank the same as this one? If so, much acclimating is not needed...but a little can never hurt.