Albon fell ill on Saturday morning and had to sit out the rest of the Italian Grand Prix weekend, with his Williams seat being taken up by reserve driver Nyck de Vries.
On Monday Williams announced that Albon suffered "complications which led to respiratory failure" after the procedure, which meant Albon had to be re-intubated and moved to the San Gerardo hospital's intensive care unit.
With his condition improving overnight, Albon was moved back to a general care ward on Sunday and is expected to be discharged from hospital on Tuesday.
"Further to Alex Albon’s diagnosis of appendicitis on the morning of Saturday 10 September, he was admitted to San Gerardo hospital for treatment. He underwent a successful laparoscopic surgery on Saturday lunchtime," Williams released in a statement.
"Following surgery, Alex suffered with unexpected post-operative anaesthetic complications which led to respiratory failure, a known but uncommon complication. He was re-intubated and transferred to intensive care for support.
"He made excellent progress overnight and was able to be removed from mechanical ventilation yesterday morning. He has now been transferred to a general ward and is expected to return home tomorrow. There were no other complications."
The team added that Albon is still hoping to be able to return at the Singapore Grand Prix on last weekend of September.
"Alex’s full focus is on recovery and preparation ahead of the Singapore Grand Prix later this month.
"Alex would like to thank the excellent care and support he has received, including from the nursing, anaesthetic and ICU teams at Osperdale San Gerardo – ASST Monza, the FIA medical team, his Performance Coach Patrick Harding, the Williams Racing team and Dr Luke Bennett from HINTSA Performance."
De Vries, who took ninth on his F1 debut in Monza, will be on standby in Singapore should Albon not be able to race.
