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Already got Certificate of Naturalization, do I have to apply for passport straight away? Is there a time limit?

I still got my original country's passport with ILR, and can use that to get in and out the country right?

A passport is a travel document. You would only need a passport if you wished to travel outside the UK as a British citizen. Millions of British people don't have a passport. My old granny never held a passport in 90 years of living but that fact did not run counter to her claim to be a British citizen.

Regardless of whether you get a British passport, the moment you are issued a naturalisation certificate, you become a British Citizen.

If you are from a country that does not allow dual nationality (such as India) then you are potentially travelling illegally with an Indian passport even though you are no more an Indian citizen

However, it is strongly advisable for a naturalised British citizen to obtain a British passport.

The passport can be used as a document for day to day evidence of British citizenship, allowing the naturalisation certificate to be kept in a safe place. Preferably not in the same place as the passport. Copies of both documents should also be kept at a different location.

This greatly reduces the risk of being unable to prove British citizenship at a critical time.