This period of uncertainty is keeping the players on their toes, according to Robinson, who is on a season-long loan from West Brom.
Caretaker-manager Chris Evans and head coach Steve Wigley got off to a winning start and secured a place in the fourth round of the FA Cup with a 4-0 victory against League Two Lincoln.
Robinson, 31, said: "We are all fighting for our shirts because you do not know who is watching.
"You have got to be professional and get on with your job. It was a great shift by the lads.
"The FA Cup is special and these ties are always difficult. When you play against lower league teams they always give you a game.
"We managed to grind them down and get the right result. We got two early goals in the second half which saw them off.
"The scoreline was a fair reflection of the game. I was more pleased with the clean sheet than the score as we have not been able to keep one in the league."
Bolton face a more formidable test on Wednesday when they return to Barclays Premier League business and a visit to Arsenal.
Robinson is relishing the challenge as they look to extend their unbeaten run to six matches.
"We know it will be tough but we have to go there and be professional," he said.
"We have to carry on like we did against Lincoln and stick up for one and other. We will be underdogs. All the pressure is on Arsenal and we will try and take the game to them."
Bolton took control of the cup tie with two goals in quick succession.
The first one in the 48th minute was slightly fortunate, however, with Moses Swaibu deflecting the ball into his own net.
The second a minute later was a clinical finish from Lee Chung-yong, who was left unmarked at the back post when Ivan Klasnic set up the opening.
Defender Gary Cahill extended Bolton's lead in the 83rd minute with his sixth goal of the season before substitute Mark Davies completed the scoring in stoppage time.