any old place
Any place at all. Typically used in negative constructions to convey that one should go to a specific location or type of location. No, you shouldn't meet your date at any old place. Make sure it's somewhere public. Oh, we can go to any old place you like—I'm just happy to spend time with you.
any old thing
Any thing at all. You can wear any old thing tonight, it's not a fancy event. It's not just any old thing—it's my grandfather's pocketwatch. We need to find it!
any old time
Any time at all. Typically used in negative constructions to convey that something should be done at a specific time. You can just come into work at any old time. You need to be here before we open to the public! I'm free all day tomorrow, so we can meet at any old time.