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Every DirectX11 compatable graphics cards can run every game at maximum settings.The thing is that not all of them can run them at max with high FPS.
And while i do agree that the HD 7770 will run games very well at that resolution, there are games that it can't even play on high @ that resolution.We're not talking ultra here.
And there is also one game that i assume it cant play with more than 40 FPS avarage even on its absolute minimum settings(Crysis 3 Alpha MP).
And now to answer the quistion.The HD5770 is a decent card but don't expect wonders.It will play most games on their high or medium settings with very satisfiyng frame rates but dont expect wonders.
I've got a HD5770 too and i used to play Skyrim @ 1080P on high settings, highly modded with constant 60 FPS.It looked better than ULTRA for the most part but it required a lot of tweaking and optimisation to get it to work this well.
Basicly you'll be able to play Skyrim on High or Ultra settings and get decent frame rates.And yes you'll be able to play League of Legends on its highest settings with a minimum of 60FPS.
(The HD 7770 is around 30% faster than a HD 5770)
BUT the HD5770 is not worth nearly that much.Currently a new one costs about 74 pounds.
http://www.scan.co.uk/products/1gb-msi-hd-6770-pcie-21-(x16)-4400mhz-gddr5-gpu-800mhz-800-cores-hdmi-d-sub-dual-link-dvi-i
For three pounds more you can buy a HD 7770 which is quite a bit faster.
http://www.amazon.co.uk/dp/B007EME3XW/?tag=pcp0f-21
But if you're looking for something a bit more serious you should look into a GTX 660 which is almost which is almost twice as powerfull as a HD7770.
http://www.overclockers.co.uk/showproduct.php?prodid=GX-174-MS&groupid=701&catid=1914&subcat=
Or if you're serious about gaming you should look into a HD 7950 which will allow you to play every game except the Witcher 2 and Crysis 3 at maximum settings at your resolution.
http://www.overclockers.co.uk/showproduct.php?prodid=GX-148-MS&groupid=701&catid=56&subcat=
It is around 25% faster than a GTX 660.