Possibility 1 - You do not mention how you installed the printer. My first step would be to delete the printer and add it back, while logged in as administrator. Make sure you follow the Intel netport print server instructions for adding a printerto an NT Client. This may be different from using the Add Printers icon (I am not familiar with the Intel netport print server, it may have its own printer setup). Do the test print and see what happens. If the test print works without pressing the start button, it should work for other print jobs.
Possibility 2 - I have had corrupt user profiles in NT 4.0 cause network printer problems. I do not know how to determine if a profile is corrupt, but try to print as the network administrator. If the printer then works properly, it does not automatically mean that the user profile is corrupt, but does indicate that the problem is related to a particular user. It is possible that all of the profiles on the NT Client are corrupt. You could create a new profile for the Guest User for testing.
You can delete the user's profile and create a new one, but first make sure there are no user files, or worse software, located under the user's profile on the NT Client. I recommend to users to store their files on the NTFS partition instead of the C: partition which is FAT in our case and also where the profiles are located.