The person’s savings increase by Rs. $2$ each day. This creates a series of daily savings: $1, 3, 5, 7, ...$

Let’s calculate the total savings day by day until we reach a value that is both a perfect square and a perfect cube.

Calculations:

1. 1st January: Savings = $1$ Total savings = $1$ ($1$ is both a perfect square and a perfect cube)

2. 2nd January: Savings = $3$ Total savings = $1 + 3 = 4$ ($4$ is a perfect square but not a perfect cube)

3. 3rd January: Savings = $5$ Total savings = $1 + 3 + 5 = 9$ ($9$ is a perfect square but not a perfect cube)

4. 4th January: Savings = $7$ Total savings = $1 + 3 + 5 + 7 = 16$ ($16$ is a perfect square but not a perfect cube)

5. 5th January: Savings = $9$ Total savings = $1 + 3 + 5 + 7 + 9 = 25$ ($25$ is a perfect square but not a perfect cube)

6. 6th January: Savings = $11$ Total savings = $1 + 3 + 5 + 7 + 9 + 11 = 36$ ($36$ is a perfect square but not a perfect cube)

7. 7th January: Savings = $13$ Total savings = $1 + 3 + 5 + 7 + 9 + 11 + 13 = 49$ ($49$ is a perfect square but not a perfect cube)

8. 8th January: Savings = $15$ Total savings = $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64$ ($64$ is both a perfect square and a perfect cube, as $64 = 8^2 = 4^3$)

The total savings amount to 64 on 8th January, 2023, which is both a perfect square and a perfect cube.