Filtering and Fourier Transforming, does the order matter? I have a signal $x(t)$. I want to find the Fourier Transform of it, $X(f)$, and then extract a narrow frequency range from $X(f)$ by use of a Band Pass Filter (BPF) in frequency domain.

Can I instead filter $x(t)$ by using a BPF in time domain and then find the Fourier Transform of the filtered signal?

I believe these two are equivalent.
There may be some slight differences due to the band-pass filtering of FFT results in the frequency domain being a circular convolution (with some wrap-around artifacts) rather than a pure linear convolution.

If you filter first, starting in time before your FFT window, any windowing artifacts from any out-of-band spectrum will be reduced before leaking into the FFT filter's pass-band.  Or alternatively, you could zero-pad the FFT window by the length of your filter's impulse response.