Ok, so I've never been in a 1on1 fight in eve where both players had every intention on fighting. My only solo experience is when Intys try to tackle me on a gate for their buddies on the other side and I insta pop them and warp off, which I wouldn't count as a true solo fight. So here is my short story.
I decided to build a Rupture to roam around lowsec and see what I could find, and when I'm 1 jump out I see a group of Hulks and an Orca on scan. So I warp to the belt and act like a retard by looting their wrecks, making sure not to MWD and just slooowwwllllyyy move from wreck to wreck. After about 2 minutes of me doing this the Orca warps off. Naturally I assume he's going to get a ship to fight me with. I give it a minute, and then burn 60k away from the hulks (All 3 hulks had t2 drones out and didn't want to be fighting against an extra 15 drones). A harbinger then warps in (I'm in a t1 cruiser vs a BC) and I'm not sure if I should just forget it or not. I keep burning away from them and then fullstop to see if the harbinger is coming toward me. Sure enough he's lurking in my direction so I turn to engage. We both lock each other at about 30k and hold lock for a short while while we continue to crawl towards eachother. (Remember I cannot be the one to fire first since we're in empire) He agresses at 20k and I immediately open fire, disrupt and launch drones at him. The fight starts off great, it appears he has close range ammo loaded since I have him down to about half shields before my shield is even dented. I set orbit to 15k and we continue to circle around eachother. He's the first to hit armor and is about 3/4 armor by the time my shields are down. Looking good. Suddenly he gets off a hammering shot and our armors are even. He was burning towards me and I was too slow to react and he webs me and each volley is taking serious chunks out of my armor. I'm at about 1/3 cap at that moment but I'm forced to MWD away from him and web him. He had me in web for about 20 seconds and my armor was seriously critical at that point with about 2/5 remaining while he sits at 1/3. Not looking good for me at this time, so I burn back out to 20k where he couldn't seem to hit me from before, and I widdle down his armor to about even with mine. I'm completely out of cap, and I'm forced to use the remaining cap to tackle him while he burns toward me and lays the final blows. We both exchange "gf" in local and a few other words, and he reports to me that he was at 62hp of structure when I popped. All in all I think I managed the fight very well except for the one instance where I let him burn at me and get me in a web which ended up being the deciding moment in the fight. I'm fairly confident that if I would have maintained range I would have won the fight handily. His char had about 2 years on mine and also a better ship, so I'm very happy with how the fight went overall besides the one bad pilot error by me.
I'm sure you guys won't find this very interesting, but it just happened about 5 minutes ago and was very very exciting for me. All in all the fight lasted about 5 minutes while his corp buddies in Hulks sat in the belt watching. A good EVE experience.