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The negative square root of nine is neither rational or irrational.
The answer to this problem is 3i, which is an imaginary number. Imaginary numbers, by definition, are neither rational or irrational.
It's a rational number. 9 is a perfect square. What you have asked is written as -sqrt(9), which can be written as -sqrt(3*3) which is - 3. That's perfectly OK. What would not be OK is something like -sqrt(10) which is irrational because it cannot be represented as a fraction or a whole number or a repeating decimal or a decimal that terminates.