In a way you've got options - which can be harder! Most Dads coming on here in this situation are desperate to get time with their baby and the Mother is refusing it and they have to go to court to get it ordered. Sometimes they've split up when the baby was very young - sometimes situations a bit like yours. But I do get that your situation is a bit different with it being a one night stand and not even dating. And you didn't choose to start a family. A lot of men never do choose to start a family - and then it happens. It's usually the women wanting children! I know married Dads who've not been sure about wanting a child - but then one comes along and they are smitten completely and highly protective of their children and adore them. But it's different being a full time Dad in a couple to having a separated parenting set up where child lives with both of you part of the time. That can have its pros as well as cons though - sharing parenting is better in a way - than in a traditional set up where the Mother is the full time carer and Dad just works hard and pays the bills and doesn't get to do much parenting (just as an example). You can do things your own way more
You get real quality time and plenty of breaks while the other parent looks after them!
You could actually see it the other way round - that the Mother is looking after your child for you so you can have breaks from childcare. I know it's not quite like that - a woman has a choice of an abortion and a man doesn't - we can't imagine having that choice for ourselves.
I think the childcare aspect could work quite well if you live with your Mum and have a bit of help now and then or someone else there if you're working on your days with the child and don't get home till later or something.
The main thing is - that nobody falls out and everyone treats everyone else with respect - even if from a distance. When things were fairly amicable with my ex, we would normally just have a handover chat on the doorstep - any info about what was happening when or any clothes that needed swapping over etc or if the baby had a vaccination appointment and who would take him. As he got older I would occasionally be invited in for a cup of tea to chat when taking him back (she had a partner by then or I wouldn't have felt comfortable about it - and didn't really anyway but went along with it so things were amicable in front of my son).