Step 2. Center it in the map.
Step 3. In your browser's location bar paste in:
Step 4. Copy the info in the popup dialog [for example: (45.483254, -122.838127)]
Step 5. Go to this page.
Step 6. In the second section ("Decimal Degrees to Degrees Minutes Seconds") type in the decimal (latitude,longitude) you got from Step 4.
Step 7. Click compute and copy the longitude from the bottom right box [for example: -122° 50' 17.2572"]
Step 8. Go to this page.
Step 9. Enter the values in the first 3 boxes [for example: 122 degrees, 50 minutes, 17 seconds]. Leave off decimal fractions and negative signs.
Step 10. If your value in #7 started with a negative, selected West. Otherwise select East. [in this example, I choose West]
Step 11. Click compute
Step 12. There you have it. Your Local Apparent Sidereal Time. In my case, it was 04:02:03.4 LST.
Oh wait, you are wondering wtf LST is? Glad you asked.
Thanks to qmchenry for providing Steps 1-4, FCC for the longitude converter, the US Naval Observatory for the Sidereal converter and to Wikipedia for the graphic.