Europe's biggest budget airline said in July that the former Malaysia Airlines boss, who denies that he is bound by such a clause, would step down at the end of the year.
But after easyJet announced Bellew's appointment as its new chief operations officer a week later, Ryanair filed legal proceedings to try to delay the move until 2021.
On the opening day of the case, a lawyer representing Ryanair listed information he said the airline could not allow to be passed to its rival, including details of delays to the delivery of Boeing's grounded 737 MAX aircraft as well the terms of deals Ryanair has signed with partners.
Delays to the 737 MAX have slowed down Ryanair's growth plans and could halt the Irish airline's expansion next summer.
"Mr Bellew has all of this information ... in his head, and it would be of immense benefit to a rival," senior counsel for Ryanair Martin Hayden told the court, adding that it would be far less relevant in a year's time.
Hayden told the court that the initial contract Bellew signed in October 2017 did not include a non-compete clause, but said a bonus scheme accepted by Bellew at the time barred him from joining a competitor for 12 months after leaving Ryanair.
Bellew received a bonus payment of 1.13 million euros ($1.25 million) in June this year, the court heard.
Bellew left his role as CEO of Malaysia Airlines two years ago to return to Ryanair, where he had been director of flight operations before leaving for Kuala Lumpur in 2014.
Tasked with tackling a pilot revolt that resulted in Ryanair's first ever strikes, Bellew was part of an effort to patch up relations with staff and agree deals on pay and conditions with trade unions that have quelled the unrest.
But by late 2018, Chief Executive Michael O'Leary told the court, there were "problems emerging on a weekly basis" and in November he sent Bellew a memo to say his performance was unacceptable.
O'Leary read out parts of a reply Bellew had sent a few days later, describing the operations chief's response the the criticism as "gobbledygook" and "waffle".
In March 2019 Bellew was told in an annual performance review that he would not be offered fresh share options.
O'Leary learned of Bellew's plans to move to easyJet only a day before the British company announced the plan, Hayden said. Correspondence between Bellew and O'Leary became "quite fulsome" after this, Hayden told the court.
Bellew has not commented publicly on the case, but his solicitor John Rogers told the court that his client felt he had been "pushed out" of Ryanair by O'Leary.
An Irish national, Bellew had been viewed in some quarters as a possible successor to O'Leary. Bellew described his return to Ryanair in 2017 as "a form of national service" to help what he described as Ireland's greatest company.
An easyJet spokeswoman declined to comment on the case, which is expected to last until the end of the week.
By Conor Humphries