It's not really possible to do that... not easily anyway. You can run a webserver on your local computer, but it's a major pain. I'd suggest editing the code on the server itself. You log in through SSH and then you can run a command line editor. It's good for small/quick changes, but if you're doing a major edit, it's probably easier just to reupload. I personally use SSHFS, which lets me mount a remote server directory via SSH as though it were a local folder. If you're not using linux or something similar I don't know where to point you though.
Now, I think I've redone a much better script in the past few hours. The actual output is ugly as sin, but that's the part I assume you're going to tweak. The code itself is a lot more functional, allowing you to upload images and descriptions, and delete things from a web interface, as well as get a quick thumbnail-ish view (it doesn't actually thumbnail, but just lays them out the same way) of all the banners. It's not perfected yet, but I'm tired tonight, and it's mostly functional. I'd like you to go play with it and find out what's still broken, and then I'll fix it tomorrow.
The display code.
http://xepher.xepher.net/Banners/bannerDB.php which doesn't have many banners installed yet, so it doesn't really show much.
The admin interface is accessed by adding "?admin=1" to the file's url.
http://xepher.xepher.net/Banners/bannerDB.php?admin=1 It's set up to ask you for a password on initial install, and after that require you to login (with cookies) in order to add or delete banners/descriptions. The current password is "slingshot" so you can try it out where it sits.
The raw code for the file is viewable at
http://xepher.xepher.net/Banners/bannerDB-source.php if you want to copy it to your own folder and start playing with it. You mainly just need to edit the "$baseurl" variable to tell it where it sits, and it should install itself from there. Note that the admin view uses a lot of fancy graphics and some javascript files that are located in my particular install. You don't need them for it to work, but if you want them, you can just download them from my install with the "save as" thing or I can make you a zip file of all of it sometime tomorrow. You can pretty easily strip out the java/pics or replace them with your own if you want. If you try your own copy and get it screwed up, delete the database file "banners.sqlite" and it will start back at the install screen and ask you for a new password.
Oh, if anyone else wants to play around with it, feel free to log in and add your own fake banners to see how it works. Let me know if you find bugs.
Hmm... one more thing. There's a couple bits of code leftover from my image catalog program. Two variables, $width and $height which are set to 400 and 200 respectively right now. If you decide to go with different sized banners, you may want to change those to match, or it could make some of the stuff look funky.