The teams that day -[You must be registered and logged in to see this link.]
From the picture it looks to me that the Bolton player is close to the goal line (foreground) and if so the Bolton player running back behind him isn't going on an overlap, so more likely getting between the ball and goal to defend it.
Assuming then we are defending and Liverpool attacking, then my guess based on players in that era sticking in their positions rather than being more fluid on the field as these days, then the Bolton player is more likely to be a defender and the Liverpool player a striker (wingers sticking mostly to the touchlines).
I'm guessing the Bolton player is therefore a centre half (Murphy or Ball) and the Liverpool player Liddell or Stubbins.
If the Bolton player is running back to goal then I assume the picture is taken to the right side of the goal area, so it seems the odds therefore if players stuck to their positions more than didn't would be Danny Murphy and Billy Liddell.
However a google at Liddell pictures seems to show he had a centre parting in his hair and the picture seems to suggest the player had a side parting - which Stubbins pictures seem to show he had - so my guess is Stubbins for the Liverpool player.
Just to complete my little analysis I tried to find a picture Danny Murphy but failed but I did find a team picture of Wanderers for 1958/9 season and it seems pretty clear to me that the Bolton player is in actual fact Roy Hartle.
Seeing he isn't shown on the team for that day (in the link above) I can only conclude that either Bonce has the wrong year for the picture or the stat site is wrong?
Bolton didn't seem to play Liverpool twice that year either in different seasons or in the cup.
Roy Hartle isn't shown as being in the team for the match the following year...[You must be registered and logged in to see this link.]
...so I suspect the picture is a few years younger than what Bonce believes it to be???
Although the stat site may equally be wrong as I've not bothered to verify its accuracy either.
Sorry, didn't set out to doubt your word Boncey, or create a mystery!