Taking someone up on something means accepting someone's offer. You can take someone up on a favor for the future, or you can take someone up on a favor that he or she has offered in the past.
For example, if a friend offers to help you with your taxes and you think you may be able to use his help in the future, you can say, "Thanks for the offer. I may take you up on that." This means: I may ask you for help in the future just as you offered.
Similarly, if your house needs painting now, and another friend offered to paint your house, you can call this friend when you're ready for his help and say "Hey man, I'm ready to take you up on your offer to paint my house. I hope the offer still stands." This means: I'm ready to accept your offer, and I hope the offer is still good.
A: So have you done your taxes yet?
B: Oh, no. I usually take care of that around late February. I don't want to think about taxes right now. What about you?
A: I worked on mine last weekend. Let me know if you need any help with yours. I can definitely help you out.
B: That's right! You're an accountant! Ok. I may just take you up on it. I'll call you late next month.
A: Alright. By the way, do you know any good house painters? I need a couple of rooms in my house painted. I could do it myself, but I don't have the time.
B: Actually, a friend of mine is a painter. He said to call him if I ever needed something painted. I can take him up on his offer and call him up.
A: That would be great. I can probably put in a couple of hours to make the job go faster.
B: Ok. If all three of us do it, we can get it done in no time.
The offer still stands - the offer is still good and valid
in no time - in very little time; in a short time
Alright folks, think of someone who offered to help you with something. Perhaps you can call them up now and ask if the offer still stands. You can say, "Can I take you up on your offer to help me with my English?" ... or something like that.