Another Nap Time Question

Yes, he needs to be left alone to fall asleep. If you allow him to stay up you are doing more than just having to deal with the crankiness. You are robbing him of rest time that helps his brain grow and develop. He will get overtired and start having more and more difficulty going to sleep. At 13 months old, he needs two naps (a morning and afternoon) - make sure you are getting him to his crib before he is too overtired. He's just a bit accustomed to fighting it, so you need to "re program" him. If it worked for him at night, it will also work in the daytime.

He needs to nap.
Sleep deprivation is not good.

Does he have a lovey?
Or a crib toy? My kids loved the Fisher Price Ocean Wonders thing.
And they had loveys. Which they used to sleep/nap and comfort with.

Also use white noise. I use a fan on low.

Make the room DARK. Just like night time.
The difference per day time naps is, it is not dark.
So make the house dark and quiet.

My kids napped 2 times a day at that age.

Sounds like you need to do for naps what you have already done for nighttime. I'd suggest using the same method. It'll be hard at first, just like it was at night, but worth it in the long run!

I had an easier time doing sleep training for naps, as I was up already, than I did for nights when I just wanted to go lay back down again! You can do it!

I would leave him in the crib. He needs the rest. It may take a while for him to realize that you're serious, but he'll learn just like he did with bedtime. But you really need to commit and actually be serious. Do not give in, or it will take even longer. Be strong!

Yes, it's okay and if he learned to do it at bedtime he can do it at nap time. Remember he is cranky and tired so it's for his own good and he'll learn quickly. Just let him know firmly but kindly that it's going to be this way, you will check on him later and put him down and shut his door. Then check on him in a bit.