<<Will a 1Gb Radeon HD7770 be able to replace a 512Mb Radeon HD 4850 which is now 4+ years old?>>
Yes, provided your system has minimum 500W (or greater) power supply with one 75W 6-pin PCI Express power connector.
<<Will it be quieter>>
Not sure but I know that the HD48xx series is quite noisy (sometimes like a jet is taking off) ,let's hope the HD7770 runs quieter.
<<and fit into my Asus P6T Deluxe motherboard?>>
Yes, it'll fit into your socket 1366 board without problem but as I said please check the PSU power output.