Neosporin can help with an infected nose piercing but you should use some hydrogen peroxide to help with the infection which would help much better.
It's not recommended to use Neosporin on a nose piercing but you can do so if you want too although the hydrogen peroxide will help prevent infection of the nose piercing much better.
After you have your nose pierced for the first time some redness and swelling can be normal and is nothing to worry about.
The redness and swelling usually goes away after a few weeks to a month but if not then see a doctor about the issue.
If you notice things such as pain, burning sensation or throbbing around the piercing site then you could have an infection and should see a doctor.
After you have your nose pierced you can change your nose ring out with a different one after the first 4 weeks.
When you do change out your nose ring to a different nose ring make sure you put the new nose ring in within a few hours.
If you wait too long to put the new nose ring in the nose piercing hole will start to close up.
So do not leave the nose ring out for longer than a day and the sooner you put the new nose ring back in the better off you'll be.
Wait too long and the nose piercing hold will close back up and then you'll have to get a new nose piercing.