Most Helpful Girls
Death row people had time to dwell on the fate they will receive. They go through the normal stages of grief until they reach acceptance. But it's normal for people to not go through the stages in the typical order.1
I think it's a pride things, they know they would die and there's no ways to escape it so as well die without making a fool of yourself.1
Most Helpful Guys
At least in the US, prisoners actually have to wait for a very long time for their death penalty. A lot of people have this naive idea that the judge speaks the sentence and two days later, the criminal is dead. That's not at all how it works. In most cases, inmates on death row live in prison for up to 20 years before they are executed. That's partially due to the fact that many of them try every legal possibility to get out of it again (that's also why the death penalty is actually extremely expensive). During these 10-20 years, the death row inmates get more than enough time to think about what's coming.
Furthermore, prisons in countries that still have the death penalty are typically very grim. This includes the US and Japan. They're not like the prisons we have in Europe where the prisoners are still treated like humans. There's a lot of violence and misery among the inmates. So, most of them are probably looking forward to die.
Oftentimes, death row inmates have had a very difficult life themselves. Their lives are basically a concatenation of shitty experiences: a miserable childhood, maybe some type of abuse, poverty, maybe social problems and loneliness, maybe an unhappy life as part of a gang, then of course the crime itself and its repercussions, the all the misery they have to go through in prison... after all that, dying probably seems almost like a lucky escape.
And besides, I'm sure many death row inmates ARE scared and nervous right before their execution. Especially since executions get botched on a regular order. Prisoners who are given the lethal injection frequently go through unspeakable and inhumane pain and suffering for long periods of time (sometimes up to an hour), something that can easily be called torture. So... knowing you might go through the same thing will scare most people, I'm sure.1
By the time they're executed, it's probably been sometime since they realized it's happening. Probably coming to terms with it. Nothing much else you can do.1