With studying pre-U and whatnot, I find that I don't have much free time to do workout any more.

So, (for example) is it okay if I do pushups on Monday, pullups on Tuesday, then pushups again on Wednesday and pullups on Thursday and so on? (This means I don't get rests days but it will only takes half the time to complete my workout.)

Or are full rest days really important? (but I thought pushups and pullups trained different muscle groups?)

Read starting strength,
read https://forum.lowyat.net/topic/371250
Maybe read Stronglifts.

My workout at the beginning was ~45 mins. Now at higher weights, it still goes to about an hour.

I guess 1 hour per workout is not too much to ask for.