re-wrapping .mov to .mp4

[Josh Ewing] "I have a 1.08 GB .mov file that I need to convert to .mp4 for upload to Vimeo."

Why? Vimeo works just fine with .mov.

[Josh Ewing] "The movie inspector in Quicktime says that the .mov file is format h264 853x40 ACC."

Were you given this file? That's a standard def 16x9 aspect ratio. HD would be better generally. Vimeo handles HD although there's a limit of one HD file per week on free accounts.

[Josh Ewing] "What process should I be using so that this won't take two days to accomplish."

Nothing. Upload the file as is. I'm not sure how it'll handle 853x480 (assume that's what you meant) as some encoders prefer 854x480. Encoders prefer numbers divisible by 16, 8, 4. 2 or 1 can result in quality loss. Encoding the file again will result in quality loss as well.

[Josh Ewing] " the file size is over the 500mb limit on our basic Vimeo account. "

Ahh, I see.
What's the exact duration (hours/minutes/seconds)?
Basically you'd want to get the bit rate as close to the 500MB limit without going over.
The other issue is, looking at Vimeo upload recommendations, they don't mention 853(or4)x480.
They do mention 640x360. We'd have to dig a bit to see how it'll handle a file at your dimension or otherwise encode to the lower frame size.

http://vimeo.com/help/compression
Choose 640×480 for 4:3 SD video, 640×360 for 16:9 SD video, and 1280×720 or 1920×1080 for HD.

[Josh Ewing] " Media Encoder gives me an estimated transcoded file size of 302 MB which is great."

Actually that might be a bit low. You could try to get it close to 500MB
http://www.pimley.net/bitrateinfo/
This would mean you'd want to get close to 3800kbps although the 2300kbps you'd get is still probably OK for Standard Def. I just try to hit closes to the max when possible. But that's just me.

[Josh Ewing] "but it's also saying that the estimated remaining time for the transcode is 34 hours, which I would prefer to avoid."

Sometimes those initial estimates change once you start encoding. Otherwise perhaps your computer is very slow. That is an unusually long time but without knowing your system specs it's hard to know.

In any case, start the encode and let it run for 15 minutes and see if the estimate drops.

It may be a settings issue as well but that's hard to know unless you post the complete settings. It could be deinteralcing when you don't need it. It might be doing a frame rate conversation that's not needed. It might be doing a multipass encode but that would to give you the best quality.

[Josh Ewing] "My computer is a new top of the line Windows 7 64 bit machine"

Not sure what "top of the line" is. I've seen a wide range of CPUs available today and some aren't suited to professional video work. An HP Z820 16 core Sand Bridge Xeon system would be top. A typical Quad Core Ivy Bridge i7 would still be good. A dual core i3 would be pretty slow though and I've seen those. GPU and RAM can also have an impact as well as hard drive speed and throughput if the computer starts using it as virtual memory.

Something would seem to be slowing your system maybe it's just settings turn up to an extreme. I've seen that with some encoders who's "Best" quality settings are painfully slow and often overkill.

That should be plenty fast.

There may be an issue with the settings or maybe some user preference corruption.
Try creating a New User Admin account. Reboot and log into the account and test the same file in that account using the same settings. Turn off an virus checkers or any other software that might run in the background.