Why didn't God forgive Saul when he asked for it? In 1 Samuel 15:24-25 (NIV):


  24 Then Saul said to Samuel, “I have sinned. I violated the Lord’s
  command and your instructions. I was afraid of the men and so I gave
  in to them. 25 Now I beg you, forgive my sin and come back with me, so
  that I may worship the Lord.”


Saul seems to have recognized his sin and repent/ask for forgiveness. However, he wasn't forgiven which can be seen in some verses later when the LORD rejected him and "the Lord regretted that he had made Saul king over Israel." (v. 35)

Does the concept of repentance/ask for forgiveness only work in New Testament time?
You appear to be reading this into the text. It doesn't say that God didn't forgive him. Samuel had to make it clear that Sauls sin cost him his crown, so he wouldn't go to worship with Saul. Once this was established, we see Samuel soften and respond to Sauls second plea to go and worship in v30. Therefore we see God forgives on a personal level, but that doesn't mean there is no price for our sin. I hope this is helpful.