It's always hard to say what's meant by the words of a song, since they're often closer to poetry than to prose. Here's what I think:
'Pick up and put on your attitude' means to assume an attitude, probably an attitude of arrogance or hubris, judging from the rest of the line. The verb 'to put on' in this case means 'to dress oneself'. He's talking about 'putting on' an attitude in the same way you'd talk about about putting on clothes. 'Attitude' in street culture means being tough or acting like you know more than you do. With clothes, before you put something on you have to 'pick it up', off the floor or off the hanger, or wherever it is. Saying it this way emphasizes the fact that the attitude has to be purposely put on. You have to make a conscious effort by picking it up first.
'Beat the streets' is the same as beating an opponent in a game. It means to win on the streets. The song is saying that nobody can 'beat the streets' and you'd better be aware of that before you assume the attitude that you know what you're doing.