Keeping the pot small OOP

[jonnyUCB]

PokerStars No-Limit Hold'em, $2 BB (4 handed) converter

UTG ($278.30)
Button ($232.45)
SB ($174.60)
Hero ($198.55)

Preflop: Hero is BB with J[img]/images/graemlins/diamond.gif[/img], K[img]/images/graemlins/spade.gif[/img].
<font color="#666666">1 fold</font>, <font color="#CC3333">Button raises to $4</font>, <font color="#666666">1 fold</font>, Hero calls $2.

Flop: ($9) K[img]/images/graemlins/heart.gif[/img], 3[img]/images/graemlins/diamond.gif[/img], 4[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">Hero bets $4</font>, Button calls $4.

Turn: ($17) 6[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(2 players)</font>
Hero checks, <font color="#CC3333">Button bets $8</font>, Hero calls $8.

River: ($33) 8[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(2 players)</font>
Hero checks, <font color="#CC3333">Button bets $35</font>, Hero calls $35.

Final Pot: $103


My turn and river play is what I'm concerned with. His preflop min-raise followed by his flatcall of the flop (after a second's pause) made me think he had aces, though a weaker king came into mind. Villian has not been that aggressive in past hands - do I fold this river or make a blocking bet? What's the best line for a hand with showdown value here?