I recently saw that someone created an Art Gallery Guide petpage:
www.neopets.com/~MayblIt sums things up pretty nicely.
You'll want to make sure you don't submit the piece too early. Usually 48-24 hours before the day of the art gallery is optimal. Otherwise it gets lost in the system, buried under all the other submissions. Not a huge time frame, unfortunately.
As for the art itself, overall I think it's good, the composition is nice, shading is honestly more than I usually do myself. Might not want to sign it with anything but your neopets username. Don't remember the exact rule, but it could be enough to be rejected. They don't want people leading other people offsite with offsite handles, but other things get caught up in there as well. I don't sign mine at all, to be safe, though in another way that's unsafe.
Did you take this with a camera? I say that because the paper color is yellowy instead of pure white, which is fairly common to see when you use a camera for your art. Scanning art is the best option. If you don't have access to a scanner, try to get the best lighting you can. Outside daylight is the best and cheapest option. Holding it near a good lamp helps too, careful to choose a lightbulb that isn't yellowy though.
Next step, whether you scan or photograph, is to adjust the colors in a photo editor.
For yours, I upped the saturation all the way, dialed back the warmth a little bit, gave it more contrast, played around with highlights and shadows slightly until it looked better. I used Photos, which is the free app that comes with windows 10. But all photo apps should come with that basic functionality. It's still the same art, but it's showcased much better.
All that I've outlined so far could be enough to get you that extra push to get in the Art Gallery, depending on your competition that week. The Art Gallery is hard to predict, even the best artists don't get in every time. And some theme days aren't as popular, and there's more opportunity for people to get in.
I consider this next advice more optional for getting into the art gallery, and more just some things that strike me about your piece that could be stronger.
The line outlining the portrait blue background could be stronger. It doesn't match the rest of your outlines on Tomos, which while sketchy, are firm and bold. The framing line meanwhile is loose and nervous, it's not very confident. I'd say to commit to your lines, all of them, to show that you are confident in your work, and the judge should be too! Not to mention, heavy lines simply show up better.
Even though the background is a simple blue oval, I'd like to see the same care and attention given to it as the rest of your piece. Namely, I want to see you use intense, solid color like you did with the fur and other parts. Even upping the contrast on the photo, the background is still washed out and not as strong, and because you can see the pencil lines, it doesn't feel purposeful.
Really though, I feel like you are
so close to being there, I'm looking at what feels like a ton of nit-picky details, but hopefully they add up to being able to get in.