As others have mentioned the only sure fire way to exactly transfer your fit is with a professional bike fit but you can get close on your own.
I just matched a fit from one bike to another bike that I just built and the measurements I took to get my position close to the original were:
-Seat height from Bottom Bracket
-Drop a Plumb line from seat nose, measure distance to center of bottom bracket.
-Measure how far from level your seat currently sits. i.e I run mine slightly nose down to where the level bubble doesn't sit in the center of the two lines but barely touches the inner line on the one side.
-measure reach from seatnose to center of bars
-run a level from seat to bars and measure drop from bottom of level to top of bars. If the seats not level it's ok, as long as you are using the same seat with the same angle then the drop to the bars will be relative.
It's important to note that i used a seat on the new bike that very similar in size as the one on the first bike. I also used one of the specialized adjustable stems which uses different shims to get different angles. I bought a 120mm and 110 mm and the shop let me return the one I did not need. This allowed me to get my fit close without guessing or buying and trying multiple stems. The shims I had allowed for +/- 4 +/-8, +/-10, +/- 12 degree angle changes. Because the bikes had different top tubes, headtubes and angles I went from a 110 -8 stem, to a 110 +12