I have 34o.
This is my second orbit, villain is new to me.
Seems aggro but also donkish.

ANTES/BLINDS
villain posts blind ($3), hero posts blind ($6).

PRE-FLOP
bunch of folds, someguy calls $6, whoever folds, villain calls $3, hero checks.

FLOP
villain checks, hero bets $15, someguy folds, villain calls $15.

TURN
villain checks, hero bets $30, villain bets $125, hero calls $95.

RIVER
villain bets $255, hero ???

At $50 to $100 NL, when the virtual nuts is one card and there's no way your opponent can be sure you don't have that card -- and he pushes -- he has it himself at least 8 times out of 10. You were in the big blind so there's no way he can be sure you don't have a 5. T5 65 or 35 would totally make sense for you.

At 3/6, I have no clue. Maybe he's playing your cards, knowing that most of the time you won't have the 5 and will have a very tough call.

I fold anyway, fwiw (not much).

I called, he showed .
Dunno if it's the right play though.

Ice, my hand is junk but it is better junk than QJ.
You say you can drop QJ easier, well I think this 34 can be dropped just as easily.

As for improving, QJ makes one pair and my hand makes two-pair or trips.
And each hand has the same number of outs (give or take, depending on suits).
I see no problem with an offsuit 4 hitting, since it's only improving a gutshot draw or hitting a random twopair.

Plus if I am called and we happen to get to showdown, I could still have the winning hand.

Anyway, I think this isn't a very big issue in this hand.