For position players you will not get any stats once you reach 162.
For outfielders you want to be at 1 on the last day, and then you will get the stats for all OF that day.
The pitching limit is not a hard stop. If you have 1-10 innings left, you want to pick up as many starters for the day you go over as you can. Plan ahead. This is a very common stategy to use, and ignorant newbies complain about it every year. Once you are over, you will not accumulate stats, even though you will see them on your team page as the games are being played. They will not count.