From what you are describing, you have either an A1 or A2 upper receiver with the built in handle. Your options are many and as varied as your choices of scopes.
You could...buy a flat top receiver with a 20" barrel, which would lend to better (lower) scope placement, too. Price of that combination is varied as well, depending on the barrel (type, manufacturer, etc.), the receiver, whether or not you are buying it as a complete package or as a barrel assembly and adding your own handguards, gas tubes, etc. But that is something that can easily be done. And with a bit of shopping, might even be done inexpensively.
Again, search back through some of the previous posts on here and you'll find links to quite a few vendors who carry just about anything you could possibly want to add to or modify on your AR.
Yes, you could also change out your existing barrel with a 20" model. Find the barrel you want, keep in mind price will vary with manufacturer, type of barrel (stainless, chrome moly, bead blasted, fluted, with or without flash hider, with or without gas tube, front sight post, gas block type, what rifling twist, etc.), length, what chamber (5.56, .223 Rem, Wylde - or any of the many other calibers available)
Along with that you might need to invest in a few tools like: an upper receiver action block to hold your receiver while you remove/install a barrel. (these run about $40, some less and are available from most of the shops that carry AR and gun parts - again see the previous posts), a good armorer's wrench - I posted a link to Midway USA for a DPMS model, a small punch to knock out the gas tube roll pin, a set of snap ring pliers - any hardware store, some non-seize grease for the receiver threads - auto and hardware stores again, and a vise. And maybe check out this link, too:
It really isn't hard to do. You'll spend more time (I hope) researching and shopping for the parts that fit your needs and wallet. Disassembly and reassembly can take minutes.
Here's a few links again to check out for barrels and parts:
I chose those because I have had experience with them in the past with good results, and based on your previous post, I take it you are trying to keep it within a budget. Don't worry, we all do. Some might have deeper pockets than others, but as long as what you have works for you is what matters most.
Good luck and keep us posted.
Oops - forgot to add...any good gunsmith or gunshop can thread your 16" barrel with no problem. It's not like you are cutting the size down below 16", just threading it for whatever attachment you want to add - flash hider, muzzle brake, suppressor, etc.