I am not sure on the resources you have at your disposal, so I will give a few answers.
In all honesty, you shouldn't be having this sort of problem, I saw a client that had 30 printers on his machine as he had a dodgy DHCP router and it kept dishing out new addresses.
Based on the fact that you said the same name, I am guessing it will be the same printer, please correct me if I am wrong.
First and quickest option, You may want to look at a static IP. This will completely stop the need for updating the IP address.
Next way which will accomplish your goal is to have a proper DNS/DHCP set up (with low TTLs). I still think the previous suggestion is better, but, if it does update the IP, at least the DNS entry will be fresh. Also if you have to use DHCP, consider using MAC reservations for devices that will be permanent/there for a long time.
Another suggestion for now is to use your hosts file. If you create an entry for
x.x.x.x printer, you should be able to set up
printer as the printer, then just update the ip here. You may need to flush your DNS ((elevated)
Command Prompt >
ipconfig /flushdns) fpr the settings to be applied.
Lastly, and this isn't really what you want, but I have to throw it in - if you are willing to update the hosts file, why not just update the IP of the printer in the printer settings page.