Can I travel outside the UAE while a bounced cheque case is open against me?

Possibly — but verify before booking anything, because a travel ban can exist without you having been personally told. In civil enforcement of a bounced cheque, the creditor can ask the execution judge to impose a travel ban on the debtor, and such orders are granted while money remains unpaid; you would typically discover it at the airport unless you check first. So check. Dubai Police offer an online service for the status of financial cases against your Emirates ID, and the Dubai Courts portal lets you search cases in your name; other emirates have equivalent court inquiry services. Because there is no single register covering every emirate, the most reliable route before international travel is a check run by a lawyer across the courts where the other side might plausibly have filed. If a case exists but no ban yet, that can change at any time while the debt is open. If a ban is in place, the practical key is the debt itself: settling, or agreeing a payment schedule the creditor accepts, supports an application to the court to lift the ban. Given your date is a month away, have a lawyer run the checks now and, if needed, open settlement talks immediately.