The correct answer is A. $x(t+4)$.
The frequency response of the system is a delayed version of the unit impulse response, with a delay of $4$ samples. This means that the output of the system is the input signal delayed by $4$ samples.
To see this, consider the following example. Let the input signal be $x(t) = u(t)$, where $u(t)$ is the unit step function. The frequency response of the system is then
$$H(f) = e^{-j4\pi f}$$
The output of the system is then
$$y(t) = \int_{-\infty}^{\infty} x(t-\tau) H(\tau) d\tau = \int_{-\infty}^{\infty} u(t-\tau) e^{-j4\pi \tau} d\tau = e^{-j4\pi t} u(t) = x(t+4)$$
Therefore, the output of the system is the input signal delayed by $4$ samples.
The other options are incorrect because they do not take into account the delay of the system.