If you need to have one instance of the data on both machines running simultaneously, then you do need NFS (well you could use SAMBA, but that would be silly). If you just want to move the data from the old machine to the new one then rsync would be a simpler option.
For example assuming you have IP connectivity between the two machines and your old machine is 192.168.0.1 and the new machine is 192.168.0.2 and you have sshd running on the old machine with root login enabled and you want to transfer everything under /home on the old machine to under home on the new machine, then as root on the new machine try something like:
rsync -avz email@example.com:/home/ /home
Note that the trailing slash after home is significant:
rsync -avz firstname.lastname@example.org:/home /home
will create another directory /home/home on your new machine, perhaps not what you want. You could alternatively push the directories from the old machine to the new one. As root on the old machine:
rsync -avz /home/ email@example.com:/home
There are lots of rsync tutorials around. Search "rsync backup examples" or similar.
Regarding down time, it might be best to run the two machines simultaneously with one instance of the data mounted NFS on the new machine. Migrate one or two pilot users to the new machine to verify that they are able to work in the new environment, then migrate the rest of the users. After all users are on the new machine, from the new machine rsync the data from the NFS partition on the new machine to a locally attached partition on the new machine, unmount the NFS partition and fix the file paths. Keep the old machine running for a week or two until it is clear that everyone can work on the new machine. Then shutdown the old machine but keep it around for a month or so before you reformat it.