Anivia is an insanely good champion when played correctly. The biggest downsides are the current top complaints by everyone right now: mana regen and mobility. Anivia is REALLY slow in movement and almost everything she does (even attacking/casting abilities/Q speed). Top this with huge mana costs for her poke/combo, and she gets to be an unpopular pick. As more and more champions become more mobile, it becomes really hard to hit for decent damage pre-6.
Another thing I almost forgot is that she also has consistently low base health (most likely due to her passive, she has the lowest starting and second lowest base health at level 18).
If you can find a good build (and manage the mana), you should be able to go GREAT with her.
Default End Build:

Rush the ROA, get boots, Morello's can help with some mana issues (Athene's for a tad help against annoying AP mids), Deathcap for a huge spike in burst, Void Staff to keep your burst high amonst beefier champs, and Hourglass to give you late game breather as well as another huge spike in damage.
Start E, Q, W, then max (R when you can) E -> Q -> W. Hitting E after a Q or R results in devastating damage and poke. I picked E as the first spell because it helps with Anivia's slow autoattacking to help last hit until level 2.
As for Spells, I personally like Ignite/Teleport. Ignite is always a very great combo finisher when you go all in (R/E/Q), and teleport can be used for two things:
- You plain died and you can get back to your lane MUCH quicker... because of her low speed. (bonus, "gank" other lanes, but eh good luck trying that slowpoke).
- You can begin casting it before your passive triggers. As long you're not moved (distorted) or slienced while still casting/channeling it, the spell will continue as an egg and you can teleport to safety while regaining your health back.
Hope that helps in regards to if she's great at scaling. (Mid game is her forte, late game she falls of a bit, but can still be a huge game-changer).
~GL