Is this a high-pass filter?
Which type of filter is it?

In the field of audio processing and tone controls it's a type of shelving filter: -

Ignore the red line - that is also a shelving filter but working at bass frequencies. Yours is equivalent to a treble shelving filter (blue line). Picture from here.
Of course, it needn't be associated with audio; it can be used in many signal processing applications. I've used one (converted into code) to counter the poor frequency response of a lame anti-alias filter in front of an ADC.