Absolutely you can do this. The above posting on potential issues - connector type (SATA, etc) you can check and can get an adpater if needed. I'm not clear from your posting if you're swapping a drive already in an external enclosure with one that's already in your case; that's what it sounds like you are doing. The hard disk physcial sizes might be different (one small and one large, for example) but you can buy an adapter to mount it in the case either way. If the one in the case is large and the external enclosure small, you'll just need a new external enclosure of the right size.
If you clone the drive, just be aware you can't plug it into your computer with the original drive in there at the same time - the clone keeps even the hardware id, so your computer will not accept two drives with the same physical address name. You'll have to reformat the drive you cloned drom (the original) if you want to use it in your system again at the same time the new (clone copy) is in use. This will cause your system to give the original drive a new hardware id so it won't conflict with the new cloned copy drive. This is probably not an issue. I would expect you will clone the drive to move your whole operating system and all data onto the larger drive, then just reformat the other drive.
One thing you can do, is you should be able to do all the steps you want - partitioning and cloning the drive over to the new drive before even installing the bigger drive, that way you can deal with any issues before taking out the known good drive you are working from now. In other words, you can confirm the new drive is setup and working correctly, before you reformat or alter your current working boot drive.
The only other thing I would add to the discussion is that if you're talking about installing a larger but older hard drive is this --> hard drives do fail and after 5+ years you might really consider if you want to risk your system on an older drive. Especially if it's a spinning drive (not SSD- solid state), and if it's ever been dropped or had errors. You can run checkdisk on it but that is a consideration. With a new drive, go for it; but with an older drive even though it's larger that may not be a wise choice in the long run (even though it's bigger so has a short term gain) if the disk fails and you lose everything with it.