I'd never accept that I was "The Last One". I'd spend the rest of my days exploring the post-apocalyptic wasteland in search of other survivors. I'd befriend an animal on the way (or if I got really desperate, a volleyball), and try as best I could to hold on to my sanity.
I wouldn't give up that I was the last person. I'd probably start writing a chronicle of the history of mankind to the best of my ability. I'd go to a library and study multiple languages, and I'd just tell the same story in as many languages as possible, and try to leave behind a big rosetta stone for the future.
While also always holding on to the hope someone else made it.
If I had ample time, I would genetically engineer a baby. Figure out a way to carry this process and pass it on to my future kids and eventually restart the population. I would have a lot of research and with no one else around ample time to figure everything out and make mistakes and not worry about being judged.
Also, since I am the only human start training the superior animals to work for me.
Reload. Just in case.
Try and find Princess Bubblegum and stay away from the Ice King
It depends. If there was still electricity, I'd probably sit and write music as long as I had power for my computer. If there was no electricity, I might walk to the city library to read. Or more than likely, I'd just have that last drink and put a bullet in my head.
I'd be genuinely shocked if I survived any war as my survival instincts aren't the greatest and I'm not particularly well adapted to surviving in the rough. I'd probably cry for a long while and then realise that I was only furthering my dehydration, which would make me even sadder. I'd probably start going scavenging and hoarding stuff, crying every time I found some semblance of a human life. I hope I'd still be in the area that I am, as lots of foods grow naturally here (very fertile) and there's lots of rain (so lots of fresh water). If there were any pharmacies left I'd probably just go source out a bunch of pain meds and kill myself, to be honest. I'd be embarrassed to be humanity's last remnant.
I don't think I'd be in a very sound state of mind, being completely alone and having lost everyone I love. I think that the only thing worth doing at that point would be cleaning up and fixing as much as I can, to make as much of the world safe for post-human life as possible. By this I mean things like shutting off power grids so that everything will be safer for the remaining life. Between all remaining houses, cars, and stores, I should be able to sustain myself long enough to make at least a bit of a difference for the world before I'm gone.
If the world were destroyed too much, though, I don't think that there would be much of a point in me lingering around just to eventually die of exposure or starvation.
How am I supposed to know what was I going to do without knowing what happened during the war? If the internet was still up I would probably try to find someone but if it was down I would probably have a large collection of the internet archived. If the war happened tomorrow, right when I woke up, and ended at the end of the day then I would probably try to find all the necessities I needed to live. So, it depends on the context of the war and how long it lasted. If during the time it happened and it seemed very grimed then I would have taken the precautions to help increases the survival later on.
I guess I would try to occupy time I had by domesticating vicious animals like wolverines, bears, mountain lions, crows, and squirrels. I would also try my hand at building electronics to make my life more convenient.
Go to a library and find some good books on survival. grow food, learn to shoot etc. I figure if I survived others did too and they'll be looking. I'm already pretty self sufficient where I am.
If you want an honest answer? I'd probably sleep for a really long time, only getting up to forage for food and water. I get depressed when I'm alone, and when I'm depressed I sleep.