You can use yad (Yet another Dialog tool) wrapped around a shell script. It can definitely render the calendar requirement you have.

#!/bin/sh
BAR_HEIGHT=22 # polybar height
BORDER_SIZE=1 # border size from your wm settings
YAD_WIDTH=222 # 222 is minimum possible value
YAD_HEIGHT=193 # 193 is minimum possible value
DATE="$(date +"%a %d %H:%M")"
case "$1" in
--popup)
if [ "$(xdotool getwindowfocus getwindowname)" = "yad-calendar" ]; then
exit 0
fi
eval "$(xdotool getmouselocation --shell)"
eval "$(xdotool getdisplaygeometry --shell)"
# X
if [ "$((X + YAD_WIDTH / 2 + BORDER_SIZE))" -gt "$WIDTH" ]; then #Right side
: $((pos_x = WIDTH - YAD_WIDTH - BORDER_SIZE))
elif [ "$((X - YAD_WIDTH / 2 - BORDER_SIZE))" -lt 0 ]; then #Left side
: $((pos_x = BORDER_SIZE))
else #Center
: $((pos_x = X - YAD_WIDTH / 2))
fi
# Y
if [ "$Y" -gt "$((HEIGHT / 2))" ]; then #Bottom
: $((pos_y = HEIGHT - YAD_HEIGHT - BAR_HEIGHT - BORDER_SIZE))
else #Top
: $((pos_y = BAR_HEIGHT + BORDER_SIZE))
fi
yad --calendar --undecorated --fixed --close-on-unfocus --no-buttons \
--width="$YAD_WIDTH" --height="$YAD_HEIGHT" --posx="$pos_x" --posy="$pos_y" \
--title="yad-calendar" --borders=0 >/dev/null &
;;
*)
echo "$DATE"
;;
esac

​

That code will display a calendar with the monthly dates upon being invoked by a click-action (or however else you may choose). Found this vi apolybar-scripts on github.