Is it neccesary to pull-up the reset pin on home made Arduino board?

Agreed. The datasheet shows a 30-to-60kOhm pull-up resistor permanently connected to the external-reset signal. This is independent of the switchable 20-to-50kOhm pull-up resistors (which in this case is disabled, presumably via the RSTDISBL fuse bit).

The datasheet also implies that the external-reset signal is directly connected to pin PC6 meaning that the reset pull-up resistor is permanently applied to this pin, even when it is working as an I/O pin (I haven't checked this yet).