Certifications are a great way to show what you know, however without a few years of experience you won’t be able to move out of the entry level position. As you are already working in help desk, that is an excellent first step. From help desk moving into a NOC position is a good next move. You won’t be able to move into a junior network admin position without a few years experience.
If the option to move into the NOC at the other company doesn’t work out, talk with your IT manager at your current company. Tell him that you are interested in moving into a network admin role, and ask him what it would take to make that happen. He’ll probably give you some decent advise. Maybe you can get yourself an internship with your current companies IT department where they can time share you with the help desk so that you can learn more about the networking side of things.
Most companies want a couple of years experience before they are willing to hire someone on into a sysadmin or net-admin position.