Im sure your organization will be alot more upset when the server decides to take a nose dive without backups or when their crucial data disapears and shows up in the competitions hands, then they would be if you drop the server for a few hours to do this correctly.

You dont want to take chances on mission critical servers. A little downtime now WILL save you a lot of downtime later. spread the work out over a few days.

Run backups one night while no one is working, maybe a weekend. Gather all the software that you will need, find all CD keys, and create backups of your configs for each program.( I swaer there is nothing more frustrating then not being able to find a disk or a cd key when trying to meet a deadline) All of this can be done and the box is still operational. Write down all the IP information, all the little settings that may only take a few minutes to look up but will bite you in the ass when you are rushing to get it back online. Then one night do a differential backup of the box to go along with the full you did over the weekend. Order you a nice pizza, and some caffinated beverage of your choice and spend your night doing the following.

Format
Reinstall
Patch
Update
Restore
Test
Go Live.

then take the next day off.