This is actually trivial unless the modern server is as bad (by todays standard) as the old one was by the standards of that time. I just did some digging and 4gb was what we bought as office systems in that timeframe - not as servers. Which means also that you don't have a large database. My oldest server is from 2007 and has 64gb memory and handles 4tb databases - something you could not do on a 32 bit OS, so your "large" databsae likely fits on a mid size SSD.
- Set up an interim server, can be as a virtual machine given how comically low powered (compared to modern hardwrae, obviously) that old server is.
- Take backup, restore on interim server. I say interim server because I am not sure you can actually restore the backup on 2016 without going to something like 2012 first - you run really old software there. You may check this, though.
- If you indeed ned an interim server, take another backup then copy to final server. THis is why the interim should be on a VM - and yes, your laptop can virtualize something as small as your old server.
Other people will easily tell you whether you need an interim server - 2008 is VERY old and may or may not be supported for a restore on 2016. I stopped using it around 2012 or so... Too long to remember compatibility for restores.
<1 day downtime is quite trivial to do - this is 2017. There is no way your old server holds a large database by modern standards - 32 bit and 4gb memory will not have allowed anything in the range of hundreds of gigabytes. Using modern tech, saving that on a SSD and then restoring it should take WAY less than an 8 hour day. Using proper storage and a direct gigabit cable (supported by the motherboard even in 2007) you can transfer around 100 megabyte per second. That is 6 gigabyte per minute - how large do you think your database is? That is not a problem at all in an 8 hour day.
There is thank heaven no way that SQL 2016 still takes 32 bit into account - which was really not current when someone at your company originally decided to go with a low end server. And yes, this is really that ancient - I have servers here that predate SQL Server 2008 and they STILL have 64gb memory and were build on 64 bit operating systems. They are end of life, though and are due for replacement. Even in 2007 - you have a lower end workstation. Which means that anything you run there is not a problem to handle with modern technology.