Usually, the phrase would be undergraduate students only need apply. This implies that the job poster only wants to receive applications from undergraduate students.
To say "need only apply" implies that undergraduate students only need to fill out an application in order to be considered. In the context of a job, this may mean that they don't need to undergo an interview. Non-undergraduate students are not expressly excluded, but they may have a more rigorous process to apply for the same job.
That being said, this is not how need only apply has always been used. In practice, it seems to be synonymous (and possibly an error in this case) with "only need apply". For example, on this gaming forum somebody posted:
Mature Casual Fun Players Need Only Apply:
We are currently recruiting mature gamers (male or female) throughout the United States West Coast to join either our Alliance or our Horde Group.
But there are also instances like this which use the phrase as would be expected from the literal analysis:
Scholarship opportunities abound; students need only apply
There is another example here of a job ad which uses the expected meaning. So it is actually difficult to discern what the ad means. It could mean that the poster only wants undergraduate students, but it could also mean the undergraduate students merely have a different process to go through.