Start with the more factual statements, such as the fact that the most recent coin is also of the highest value. This means that the ten-rupee coin was minted in 2012. Also, since the second-oldest coin is worth two rupees, the 1976 coin has a value of two rupees.
The coin in the centre is of the lowest value, so the one-rupee coin has a position of 4 (from the left, which is how I will define position from here onwards). Since there are two pairs of years using the same digits (1967 and 1976; 1989 and 1998) these must have positions 2, 3, 4 and 5, though we do not yet know the order.
Since the coin with position 2 was minted in 1989, the fourth coin, worth 1 rupee, was minted in 1998.
Because 1991 and 2002 are palindromic, their corresponding coins must be on the far left and far right (not necessarily respectively).
As the oldest coin is third from the right, position 5 has a year of 1967, and so position 3 has a year of 1976.
So far, all positions apart from 6 have been filled, and all years apart from 2012 have a place. Therefore, the coin minted in 2012, worth 10 rupees, has position 6.
The two five-rupee coins are next to each other, and the only two consecutive positions without a value are 1 and 2. Now, the two remaining values are both 2, so these can be filled in.
Finally, the coin on the immediate left of position 7 was minted in 2012. Of 1991 and 2002, only 1991 is more than ten years older than the 2012 coin. Fill in 2002 on the extreme left, and 1991 on the extreme right, and you’re done! The final solution is: