The method we all use to locate the bridge correctly is to play the harmonic at the twelfth fret and compare it to the fretted note at the twelfth. Here's how it's done:
When you just barely touch the string exactly over the twelfth fret and pluck it with your other hand, you force the string to vibrate exactly in half, producing a note an octave above that to which the string is tuned.
Pressing down at the twelfth fret should give exactly the same note. If the fretted note is sharp, then the string must be lengthened by moving the bridge toward the tailpiece.
By testing the highest and lowest strings this way, you can get the bridge into just about the right place. If you have a critical ear, you might spend quite a bit of time getting the bridge where you want it. As the strings age, or if you change gauge, you might find you'll need to readjust the bridge position.
Don't forget to check out the article on maintaining your gears. Mandolins usually have open gears which need just a bit of care and feeding.
Let's finish this job.
I pass the string through the hole in the tuner post, leaving a bit of slack over the neck:
Then I bring the free end around opposite the direction the post will turn when I tune it up. I bring the string down the center of the peghead and around under itself:
Lifting up on the free end of the string
I bend it over itself, and press downward:
More