I need to write a C program that launches another program with a modified priority, much as as the nice command does. In order to do that, I would like to find the PID of a process given as an argument (how can I do that?) and modify its priority level (how can I do that?).
Example: The command line might be ./a.out 5 sleep 500 &, and this should produce the same effect as nice -n 5 sleep 500&.
You are focusing on the wrong thing and therefore approaching the problem with the wrong idea. The key requirement is that your program must execute a specified command. Focusing on that will lead you toward how to achieve the process priority goal, at least by helping you frame the question more usefully. For example, you don't need to find any PID, because you don't need to adjust the niceness of an arbitrary process.
So how do you programmatically launch another command? The typical way would be to use one of the functions from the exec family. Since the program name and arguments are comming from the command line, execvp() is probably your best choice.
If you read their docs, you will find that the exec functions replace the process image in the current process. That is, they make the process in which they are called start and run a different program in place of the one it was running before. If the command you're going to launch will run in the current process, then it's the current process whose niceness you want to adjust, and for that there is nice().
You shouldn't need much more than those two functions and a little command-line parsing. Do read those functions' documentation carefully, however, especially execvp()'s, to make sure you set up the arguments correctly.