I am not sure what your boyfriend's work/sleep situation is. Is he working longs hours and then sleeping soundly at night, which would be expected, and that is a difficult pattern for him to change? Or is he falling asleep in the middle of the day and then difficult to arouse from a nap? I am going to guess the nighttime one.
If you two live together and you have only had the baby home for two months, it may take him longer to get used to waking at night to care for the baby, especially if he is working long hours and has been used to using this time to restore. I am not saying that he shouldn't help at night, but that you may have to help him learn to wake up from a deep sleep in a more compassionate way (no more water splashing, okay?) and mentor him with baby care.
Spell out what you need him to do: go get the baby, change the diaper, prepare the bottle and how to feed, burp and settle back to sleep. You may have to wake your boyfriend each time and also get up with him to practice these tasks, so that you know he is able to do them solo to your liking, so then you can relax and sleep through a night feeding every now and then, too!
If he is falling asleep in the daytime, I am not sure what to say. Are you concerned that he is under the influence of something that would make him sleep in the daytime and therefore not safe to spend waking time with your infant?
Go with your gut. If you suspect for a minute that your child will not be properly cared for, talk with him and work out a plan to build trust and so that he can become a loving caregiver-daddy to his son. Then, start with 10 minute walks around the neighborhood with a cell phone. (Once you feel confident he is ready to do this solo!)