How do you handle multiple urgent requests at the same time?

Give a triage framework rather than saying you work hard.

• Assess impact and urgency quickly. How many people are affected, how critical is the system to the business, and is the situation deteriorating. A whole department unable to work outranks one person's individual issue, regardless of who escalated loudest.
• Security incidents go first. Delay compounds the damage.
• Check whether they are related. Several simultaneous reports are frequently one underlying failure, and recognising that turns three problems into one.
• Communicate to everyone waiting. A realistic timeframe stops people chasing, which itself consumes time. Silence produces escalation.
• Delegate and escalate early. Involving someone else while there is still time is far better than at the point of failure.
• Apply a workaround to buy time where one exists, so people can work while you fix the cause properly.

Note: Being clear that priority is set by business impact rather than by seniority of the person asking — while handling that diplomatically — is what interviewers are checking for. So is recognising when to say you cannot do everything and ask which should wait.