My Galaxy i5503 (Android 2.2) is dead, they say it needs new software? My phone is not switching on. It's a Samsung Galaxy i5503/Galaxy 5. It started automatic switch offs even on a little pressure and now it's not getting on.

Someone checked it and said they need to reinstall the software (probably the operating system, which is Android 2.2).

My question is: Can I do it manually? If yes, then how? And, if I opt for getting it done from someone, What data will be erased? I cannot afford to lose the data of my applications (like Colornote's notes).
I have bad news:
It almost impossible to get data from your internal storage if your mobile does not turn on.
First we need to know more about the problem:

What do you mean under not turn on?

Does it light up or no signal at all? What if you plug it into the charger? Can you see any lights flash up? Can you get into recovery?

If none of the above happen, then I need to ask: Is your important data stored on your SD card? Most apps store their user data on SD Card so you can access them if you pull out your SD card and read from that.

If you re-install android, you WILL LOSE your internal data so if you have lots of important things on your phone (not your SD Card) you should not reinstall without backup

Colornote does not store your data internally but uploads to the cloud. See Their FAQ