No, it wont prevent you from doing anything. I recently did this myself.

...
I still have the old-school Minecraft account, not a Mojang account (hey, if it works, don't fix it; and it's not like I'm ever going to play Scrolls).

But I was tweaking my skin, and the app I used insists that one use a Mojang account.

So, if I migrate my old Minecraft.net account to a Mojang account, will that prevent me from logging onto the server, as a name change would?

As others have suggested if you migrate your account it won't make a difference to your ability to connect to Noobstown.com. The majority of our players now use a Mojang account to connect. The only difference is in the client you use your email to login rather than your Minecraft account name.

Be aware however currently if you change your Minecraft account username you will be blocked from accessing the server though.