Is it possible to set the Now Time in milliseconds?

I don't think it's possible. You can do SMPTE time, but it won't give the kind of resolution you want.
 
If you need to go to specific points repeatedly, you can insert a marker so you can jump to it.

slartabartfast
You should be able to set the Now Time in the Transport View to read in milliseconds (or samples which will be the effective limit of your resolution) by clicking the time display to cycle through the measurement options.

Yes, but he wants to be able to enter a specific number and have the now time jump to that place. AFAIK you can do that with SMPTE and M:B:T, but not milliseconds.

Hey, that works quite simply. Great workaround. It would still be good to have an option to show (and therefore enter) all times in milliseconds (like you can with SMPTE) but for occasional use, sorted! Thanks.

slartabartfast
The Clip Inspector properties should allow you to set a start time for the clip in milliseconds, so you might be able to kludge this by creating a dummy clip in a dummy track, set the Start Time for the clip to move it to the location you want and jump the Now Time back to the start time of that clip and then use that Now Time in the track you want to edit. 

That's a really smart workaround, and checking it out made me aware that you can actually specify clip start resolution in microseconds.
 
Here's a further refinement: Set the Nudge time to 1 ms. While observing the Clip Properties Start Time in the Inspector, you can see the start time change with each press of the nudge key. You can also hold down a nudge key but the start time won't update until you release the key.
 
Another refinement: In the Snap to Grid preferences, choose Clips for landmarks and enable the Landmark button in the Control Bar. Set the musical snap value to something big (e.g., Whole Note) and if you click in the vicinity of the clip start, the now time will snap to it.
 
Finally, it can be helpful to set the Snap value to 0.001 seconds when trying to place the now time exactly. 
 
That's one of the things I love about this forum...I never would have thought of any of these refinements without the original query and slartabartfast's clever workaround. Now someone will probably come up with refinements on my refinements...