Convolution effects width of the signal?
If $y(t)$ is the signal resulting from the convolution of $x_1(t)$ with $x_2(t)$ then it will have the same bandwidth as $x_1(t)$. $x_1(t)$ has (presumably one-sided) frequency support in $[0, 500]$ Hz and $x_2(t)$ has frequency support in $[0, 1000]$ Hz. So, when performing pointwise multiplication in the frequency domain, the higher frequency components in $x_2(t)$ between $500$ Hz and $1000$ Hz will be multiplied by zero (the value of $X_1(f)$ at those frequencies). The Nyquist frequency required to capture $y(t)$ without distortion should be any frequency greater than $1000$ Hz (twice the highest frequency in the signal).