I think these are two very separate issues. There's no reason why 4 days a month should be all you are allowed, unless there is something you're not sharing. Your DS has a right to spend equal time with both of you, as long as that is not disruptive to his sleep/education etc. so I would start with that if I were you.
Regarding the new partner, on MN there seems to be a big divide between those saying new partners shouldn't be introduced until the relationship is solid and long-lasting and those who believe that the dynamics are so drastically changed once DCs are involved that it is useful to see how things work before becoming too deeply involved.
For a lot of blended families it is not until they actually move in together that the cracks are revealed, so however long they wait, they won't know that this is 'forever' until they all try living together.
My DCs met my DP after a few weeks and it became apparent that they all got on really well, however after a couple of years we had some big issues and now after 4 years we still don't live together. I imagine that the emotional upheaval of us splitting up would be much bigger now than at the start, so I don't really understand what people are protecting their DCs from by not introducing them sooner. Relationships can fail at any point and if it's someone the DCs have become attached to, it doesn't matter if they met after 2 weeks or 2 years.
As for new man's suitability to be around DCs, you would be falling into the trap that women are often accused of - presuming all men are potential child molesters - if you made a fuss about this. Perhaps presume the best of him, ask if you can meet him to suss him out and make sure your DS feels free to talk to you about what goes on at his mum's (without you prying!)
Just because you don't want to be with your ex any more, it doesn't mean that this new man has something wrong with him to want to be with her. He is probably a nice guy and hopefully he can be someone your DS looks up to. He won't ever be his dad, but no reason why your DS can't have other male role models in his life. I was really pleased when my ex got a new GF, who was someone my DCs really liked.
Unfortunately they have since split and he has met someone else, who may also be on the way out
but it's his right to introduce them to whoever he has round. He introduces them to various non-romantic friends too, so it's not unusual for them to meet new people who may or may not be around a lot.