How can you tell from the story that the policeman is Jimmy in "After Twenty Years"?

Jimmy's letter shows he was the policeman who conversed with Bob in the beginning of the story.

In the letter, Jimmy mentions he was on time to meet Bob. While speaking with him, however, he recognized Bob as a criminal wanted by Chicago cops. Jimmy asks a plainclothes officer to impersonate him because he can't bring himself to arrest a man he used to call a friend.

I didn’t want to arrest you myself. So I went and got another cop and sent him to do the job.

The contents of the letter make it clear Jimmy was the police officer. As we look back at the story, the letter explains why Jimmy asked how long Bob was willing to wait:

“I’ll go on my way,” he said. “I hope your friend comes all right. If he isn’t here at ten, are you going to leave?” “I am not!” said the other. “I’ll wait half an hour, at least. If Jimmy is alive on earth, he’ll be here by that time. Good night, officer.”

Based on Bob's answer, Jimmy may have concluded half an hour was plenty of time for another officer to impersonate him and meet with Bob.

Bob's answer also shows he still has some affection for his old friend, Jimmy. The men may not be close now, but past memories and shared experiences are often difficult to forget. This is perhaps the reason Jimmy sends another officer to do his job.