Is it valid to perform the prostration of forgetfulness a short while after the prayer for someone who was unaware of the place of the takbir, so he said the takbir after rising from prostration to the second rak'ah while he was upright?

What this person did is correct according to the Hanbali school of thought, because the place for the takbirat al-intiqal (transition takbirs) is between the two rukns (pillars). So, whoever pronounces the takbir while standing will not have it counted for him, and he must perform the prostration of forgetfulness (sujud al-sahw) if this was due to forgetfulness or ignorance. The prostration of forgetfulness is waived if a long time has passed.