I had the same problem about 2 weeks ago. And mine is practically brand new. Got it in October of last year, use it a few times in the winter but not a lot. Come March 1st I go to a job to get started and the thing is hard to pull like yours. Plus gas starts leaking out of the carb, but I did get in started and it ran like a champ. Next Day, same thing, so I took it to my repair guy and he cleaned the carb and I picked it up he asked me how I shut it off, I told him using the switch, he ten told me to ALWAYS shut the machine off, when I am done for the day, using the gas shut off, not the switch. Mine is a 13hp Honda too. I looked at the guy like he was nut's as in 20 plus years fo using theses machines I have never had to shut of a machine using the gas shut off. But sure enough, I havent had a problem since.
He said the needle valve swells if gas is left in the carb. Again, I have never heard of this, but shutting it off using the gas valve works. Also takes a lot of getting used to. Some days I forget, and the next it's hard to start, but as long as I remember it starts first pul every morning, as long as I remember to turn the gas back ON that is.