Can being poked in eye cause runny nose?

yes. a poke in the eye stimulates tear formation. tears may run out of your eyes but most drain through the naso-lacrimal canal into the nose. you can see the start of you naso lacrimal canal if you look closely at your lower eylid in the mirror - pull the eyelid down a little and you can see a little whitish blob with a pink dimple in it at the inner aspect of the lower lid: taht's the start of the canal.

so after a poke in the eye you will get a nose full of tears on the same side as the poke (and possibly on both sides since the poke may cause teears on both sides).