How long do hornet stings hurt?

There is no delay. As the sting penetrates it feels like a fine hot needle. The effect of the venom is almost immediate. The exact time varies depending on how much venom is injected, and on how the individual reacts to the venom, but the initial pain usually only lasts for about an hour, though for a few people it can be longer. The sting site is then sensitive for the rest of the day. A small percentage of people react badly to the venom and for them the symptoms may last longer.
It is probably because the initial pain only lasts for a fairly short time that so many household remedies are claimed to cure bee stings. In fact, for most of them by the time you apply the remedy and give it some time to 'take effect' the pain would be going anyway.
The most important thing when stung is to get the sting out as quickly as possible. This minimizes the amount of venom injected. Do not squeeze the sting, but scrape it out with a thumbnail or something like the back of a knife.
Beekeepers, for whom getting stung is more common, will tell you that the more often you are stung, the less each sting hurts because the body becomes acclimatized to the venom. Old-time beekeepers hardly notice being stung.