Does anyone know what is the nucleus program like?
5 Year structured program, different tracks undergo different training, SWE/Data analyst will go for masters certification under NUS ISS, the other tracks lan lan suck thumb no masters.

Do you have to study and take exam while working at the same time?
Yes, but 1 year masters curriculum spread over 5 years, pretty sure its not hard to attain the cert. And no you cannot complete the masters in less than 5 years even if you very fast, its gated by HR.

Is this a good place for fresh graduate from computer science to gain experience and exposure?
Gain experience and exposure definitely, you will be thrown to projects for OJT, don't mistake OJT for not delivering anything, you are a full time staff under nucleus, have to deliver or your performance grading will be crap. Also, starting pay of 4.8k applies to all nucleus, a fresh graduate from some 1 year private uni marketing degree will get the same pay and progression as a fresh graduate from NUS computer science with second upper.

Is it possible to just focus on the work rather than courses to attain the certificate?
No such thing. If can just use work experience get certificate without going through tests or exams then that cert is useless by default.