According to http://forum.wordreference.com/showthread.php?t=738701 :
"appena" and "non appena" can often be used interchangeably.
In this case not only does "non" have no negative meaning but it actually acts as an intensifier.
I figured if this was supposed to be a negative, the "non" would have come before the "venni". I translated it literally as something like, "I came not just as soon as I could" and then figured the "not" part wasn't meant to be literal. I feel like dialectal/slang English does stuff like this too. I can't think of a great example off the top of my head, but think about how people might say things like, "I was there just barely two seconds when..." and "I wasn't there just barely two seconds when..." and how they both mean the same thing in colloquial dialectal English.
Different question: Why is this using remote past? The use of "came" shows that the speaker is still there, in the same place, so it couldn't have been very long ago at all. Is this the literary use of passato remoto? And yet it would only be used in dialogue. Still, this is the very first example, so maybe I'll figure it out soon...
Well, it's the Passato Remoto module.
This tense is often used in literature and story-telling. Out of context you can't tell without a whole story. "Twenty years ago, our house burned down. They called me at work. I came as soon as I could but it was too late. Everything was lost."
Clearly the person is NOT still there and it happened long ago. Most of this module will be without context. And, I think, most of my example sentence would be in the remote past tense. Hope that helps.
"I came as soon as I could" does not work in your sentence. Your sentence is "I got there as soon as I could" or "I went as soon as I could." "Came" implies you are already there as in "I got here as soon as I could." To use "came" that way you would need to be recounting a conversation in the past and not events in the past. You can use it to tell what others have done in your example: twenty years ago . . . my sister arrived two days later "I came as soon as I could". But in remote past "came" will be in conversation.
It doesn't follow the pattern for other verbs ending with -IRE, so I expect it's an irregular one, like fare, dire, bere, dare, stare etc.
Regular verbs ending with -IRE are e.g:
Here's a link to a very good, Italian webb page that explains the Passato remoto very clearly: https://www.italianochefatica.it/it/passato-remoto/