How to measure the altitude and azimuth of a star?
For the second question: If you know the coordinates of the Zenith, your latitude is exactly Zenith's declination. For your longitude you can not rely on the Zenith: the same star will be at the Zenith at the same sidereous time for every place in the same latitude, so you need a clock besides the telescope. cf. http://en.wikipedia.org/wiki/History_of_longitude#Problem_of_longitude

For the first question: The altitude of a star that is crossing the local meridian is also quite easy.

For Northern Hemisphere, a star that is crossing the North local meridian (that is, between the Zenith and the North Horizon) the latitude is
$90-alt=dec-lat$

For Northern Hemisphere, a star that is crossing the South local meridian (that is, between the Zenith and the South Horizon) the latitude is
$90-alt=lat-dec$

Same goes for Southern Hemisphere if you change also North Horizon and South Horizon.

You did not asked specifically for the Azimuth, but again you need a clock for that.