Abort an EventHandler?

Shortly, use a SelectorPane and hide EventHandler as soon as the condition is met:

pt = {0, .05};

pane = "dynamic";

graphics = Framed[Graphics[
   {{Red, Disk[Dynamic[pt], .1]}}, PlotRange -> 1, Axes -> True]
];


eh = EventHandler[#,
    {
     "LeftArrowKeyDown" :> (pt -= {0.05, 0}; check@pt),
     "RightArrowKeyDown" :> (pt += {0.05, 0}; check@pt),
     "UpArrowKeyDown" :> (pt += {0, 0.05}; check@pt),
     "DownArrowKeyDown" :> (pt -= {0, 0.05}; check@pt)}
    ] &;


   (* Norm will be better as there some subtleties with =={0,0}*)
check[pts_] := If[Norm[pts] < .01, pane = "static"; FinishDynamic[];];

Column[{
  PaneSelector[
   {"dynamic" -> eh@graphics, "static" -> graphics},
   Dynamic@pane

   ],
  Dynamic@pane}
 ]

There are a couple of issues.

1. The floating-point arithmetic could result in a point close to (0, 0) without being equal to (0, 0). Use Chop.
2. The Abort happens after pt is updated in EventHandler . So sequence of events is
   • Press key.
   • Update to (0, 0) (actually, really close to (0, 0)).
   • Not (0, 0), so continue.
   • Press key again.
   • Update to not (0, 0).
   • Not (0, 0), so continue.

You need move zero check to the start of the EventHandler and use Chop post arithmetic to succeed in the check.

pt = {0.5, 0.5};
EventHandler[
 Framed[Graphics[{{Red, Disk[Dynamic[pt], .1]}}, PlotRange -> 1, 
   Axes -> True]], {
  "LeftArrowKeyDown" :> (If[pt == {0, 0}, Print["Move on"]; Abort[]]; 
     pt -= {0.05, 0}; pt = Chop /@ pt; ),
  "RightArrowKeyDown" :> (If[pt == {0, 0}, Print["Move on"]; Abort[]]; 
     pt += {0.05, 0}; pt = Chop /@ pt;),
  "UpArrowKeyDown" :> (If[pt == {0, 0}, Print["Move on"]; Abort[]]; 
     pt += {0, 0.05}; pt = Chop /@ pt; ),
  "DownArrowKeyDown" :> (If[pt == {0, 0}, Print["Move on"]; Abort[]]; 
     pt -= {0, 0.05}; pt = Chop /@ pt;)}]

Hope this helps.