It varies with the model but on the TV you go to some menu to check for updates. I go the old school route and check the Samsung site for my model.

Be aware that I never tell folk how to update since there is a chance if the user forces an incorrect version to be installed the repair bill is high.

-> Can you tell more about the long turn on time? Is it power cycling? Or just sitting there?

The power cycling is hard on many boards so if you don't fix it, it could make the repair more expensive later.

The LN46A63M1FXZA does not find TVs at google.com or Samsung.com here.
When I search at Samsung the only result is https://www.samsung.com/us/support/owners/product/2008lcd-tv-a630?redir=LN46A63M1FXZA which could be wrong.

Is there a typo here?