A filter for for a realtime system?
For an EEG signal, I suppose you need to keep the shape of the signal. Therefore the filter should have linear phase, so the reasonable way to go is a FIR filter.

If shape (transients) don't matter much for your application, then an IIR filter would be preferable since it will give less end-to-end delay in general.

Note that suprressing 0 - 1 Hz, inevitably gives a group delay of at least 1 second, depending on how sharp the transition should be, so your real-time requirements may conflict with the start frequency.

You may try some filter design tools (e.g. from matlab) to find a reasonable compromise between frequency response and group delay.