fedupwiththis you are totally being reasonable and fair and he sounds like a massive pain in the arse!
Lots of things about your words are not really comfortable to read!
"He just gets so angry at me now" So since you had a baby he has started being angry. Was he angry before? Is he violent or threatening?
My guess is that maybe at some point you will end up parenting alone because he seems to be an angry man who thinks looking after the baby is your job.
"He basically said that I shouldn't make any plans until after baby is asleep at 7, but he knows that means I will just have to stay in. He isn't about sharing it. He literally wants to do an hour childcare a day, 7 days a week." So that's not going to work for long is it. Does he not want to be with his son?
Does he not realise the baby is only going to develop a relationship with him if he spends time with baby and does things for him.
"when he's also having to work late in the evenings"
Does he have to work evenings and weekends? Is it his own business, do you need the money? Or does he choose to do this because he prefers it to being with his son or being with you, OP?
"Doesn't seem very fair." because it isn't fair. Seems like his interest in fairness is about the same as his interest in caring for his son.
"When I check on him he is watching a dvd, gaming or doing exercise." So being in his office is actually just a place to do whatever he wants, not necessarily work. He is taking the piss. Did he want to be a dad?
Shouldn't that read ... His office is at the end of the garden, so baby isn't there stressing him out while he is watching a dvd, gaming or doing exercise!
Sounds like you need to overhaul your relationship before this lazy, angry man makes your life a misery.