Can the transient response of a system be calculated using FFT and inverse FFT?
First theoretical mistake to point out is trying to apply FFT (Fast Fourier Transform) to a continuous-time signal, since it would only make sense for discrete-time. If you actually mean just FT (Fourier Transform), then you'll be digging yourself into a hole, since the FT for a ramp is not very pretty: wikipedia link.

Instead, you should be thinking Laplace. The Laplace transform \$X(s)\$ for a ramp \$x(t) = t\$ is:

$$
X(s) = \frac{1}{s^2}
$$

The Laplace transform \$Y(s)\$ of the output \$y(t)\$ will be:

$$
Y(s) = G(s)X(s) = \frac{1}{(s^2+1)s^2}
$$

The Laplace inverse \$y(t)\$ can then be given by partial fractions:

$$
Y(s) = \frac{1}{s^2} - \frac{1}{s^2+1}
$$

$$
y(t) = t-\sin(t)
$$