How to do sink plumbing in the basement?
Your last pic seems to indicate where the drain exits to the street/septic system. If your sink bowl is enough above that point that you can run a drain line from the sink, with the proper 1/4&quot; per foot slope, and tie it into this drain, you should be fine. If not, then you'll need some sort of lift pump to get the waste water up to that level.
If you're going to end up going with the lift pump, you might consider putting in a bathroom nearby to get more use out of the pump.