1) The story of Srimad Ramayana ends with Sri Rama's coronation as the King of Kosala Kingdom. Sage Valmiki describes in the concluding chapter of Yuddha Kanda that having enjoyed the kingship for ten thousand years, Sri Rama performed a hundred horse-sacrifices. It was described very briefly about the happy life of the people of Kosala kingdom during the reign of Sri Rama.
सर्वे लक्षणसम्पन्नाः सर्वे धर्मपरायणाः ||
दशवर्षसहस्राणि रामो राज्यमकारयत् | (Yuddha Kanda 128 Sarga 106 Sloka)
2) Srimad Ramayana was written much earlier to Mahabharata. In the 272-289 Sections of Vana Parva
of Mahabharata, the story of Sri Rama was narrated to Yuddhistara by
Sage Markandeya. Though the story contains minor variations compared
to the story told in the Srimad Ramayana, those episodes describe the
story of Sri Rama in full.
3) According to the Scientific dating conducted by many scholars, The
Great War of Mahabharat between the Pandavas and the Kauravas happened
in 3139 BC. The Pandavas, after winning the Mahabharat war, ruled
Hastinapur for 36 years and 8 months until the beginning of kaliyug in
3102 BC.
4) As the story of Sri Rama was narrated in full to Yuddhisthira in Mahabharata, we can safely conclude that Srimad Ramayana was written much earlier to Mahabharata.
5) According to Srimad Ramayana, Sri Rama ruled Kosala Kingdom for 10,000 years. Please do not be cynical as to the description of 10,000 years of ruling. We do not know exact age and the longevity of people of that age.
6) If we take 3102 BC as the basis for the Mahabharata dating, then at least 10,000 years are to be added, assuming that Mahabharata immediately followed Sri Rama, to 3102 BC for getting the age of Sri Rama. Thus, we will get 13,102 BC. That means atleast 15,000 years back Srimad Ramayana might have written.
This was the available calculation.