Play the selected song that you are attempting to keep up with. Listen to the song up to five times before you try to recite lyrics. Stop the beat and practice the lyrics out loud until you can say them without stuttering or making mistakes.
Focus on the last word in each sentence or bar and say the word over and over. Play the rap beat and repeat the last word of each sentence so you time up with the beat. Speak the remainder of the lyrics leading up to the last word and determine which words will work and which ones won't. Too many words in a sentence can effect the timing and ability to keep up with the beat.
Use a metronome to keep up with the beat by setting the metronome to the same tempo as the beat. Most rap beats go in measures of 4/4, meaning four beats per line, so set the correct tempo. Practice the lyrics to the beat of the metronome and keep going even if you fumble around words or pause.
Experiment by speaking the lyrics slowly and then fast to the selected beat. You will learn how to manipulate the way you say words when you do this. Pace yourself by saying one line at a time until you remember all of the lyrics. If you're familiar with the lyrics, you're more likely to achieve a smooth-flowing transition from line to line.