They said my kidney stone is close to my bladder how long will it take for me to urinate it out? & is it normal for urine to burn at end?

Kidney stones. A majority of kidney stones will pass on their own. If you were told that it is close to your bladder, it means it has already traveled most of the way out. So it should be just another day or two. In the meantime to have some burning with urination is normal. If the stone does not pass after 2 weeks, you may need to see a specialist for assistance.
Depends... On the size of the stone. Smaller stones usually pass more quickly, but that is not always the case. Burning at the end of urination or pain at the tip of the penis or clitoral head usually indicates a stone trying to pass through the wall of the bladder. See your doctor for an alpha-blocker if you haven't been prescribed one yet, it may help you pass the stone more easily.