FIR Filter - Low Pass Filter coefficients to High Pass filter coefficients
HINT:

It looks like they shifted the frequency response by half the sampling frequency, i.e.,

$$H_{HP}\left(e^{j\omega}\right)=H_{LP}\big(e^{j(\omega-\pi)}\big)\tag{1}$$

Frequency shifting corresponds to modulation (multiplication) in the time domain. Now you just have to figure out the modulation sequence that achieves the correct frequency shift.