Suggestions for Unexpectedly Quick Promotion?

BLUF: I'm being promoted to Director after only 3.5 months in the manager role on one of the Director's teams with 6.5 years on the team and with the Company. Some business partners have not worked with me at all and are nervous about the change and at least one other manager was interested in the role, but was passed over. I'm looking for advice about how to best handle those situations.

Longer explanation: I work for a major Subsidiary of a global Fortune 100 company. My Director left the Company unexpectedly. I thought I knew what the VP would do to fill her role and was completely surprised when she asked me to take the role instead. She put me in the role on an interim basis To give me time to make sure I would like it. I took a couple of weeks to get the lay of the land and decided to accept (some approvers were out off the office). All that's left is some paperwork for Monday, but the VP introduced me to colleagues from other companies as "her new director" last week. (Note I have had previous manager jobs. I was just hired initially at this company as a department of one, that has since expanded to 7)

Thankfully, everyone on the team I was previously managing is thrilled. They nominated me (and I'm a current finalist for) Employee of the Year. All of the other Directors reporting to my VP are happy with her choice. The CFO and Chief Actuary didn't even give her a chance to explain any of her list of reasons why she thought I was the right choice for the role because they were fine with it.

I'm missing certifications I will need and need to obtain them by 12/31, so I have a lot of information to study to pass the the exams along with classes that will put me out of the office. I'm also pretty unfamiliar with the second team reporting to me and they have had a lot of turnover and some extremely challenging objectives to meet this year with a lot of pressure from the parent company. Additionally, my previous team has some new objectives with some hard decisions to make about priorities. Other challenges are that the two people the VP would like to manage those teams have never managed people before. I've sat down with the VP and asked her a ton of questions while we were out of town. I'm feeling a little bit stressed about the sheer number of things I need to learn quickly to execute well. That's probably why I'm posting this instead of sleeping.

To handle that, I currently have one on ones scheduled with the interim managers for longer than the standard time. I have skip level one on ones scheduled with everyone else and have met with each person at least once. I'm studying for my test everyday for an hour or more. One of the interim managers is extremely self motivated and I can toss her some requests to help identify lower priority items from our report with a priority list from the VP and CFO.

All that aside, there are two primary business areas that are most nervous about the change. Our operations area is heavily dependent on my new team (with the turnover) to help educate them, but the old Director was enabling operations instead of coaching them to make their own decisions. That team has 3.5 months to finish a huge project and is afraid and a little helpless. They have to finish it for my team to deliver its objectives and for them to meet theirs. The COO is concerned about what this change will mean for her. How do I best help with this? I've only met her twice, once in the employee of the year interview.

One of the higher levelmarnier leaders of our distribution area is also very concerned and has only met me once. To help make the head of that area feel like he would get to know me, my VP asked him to mentor me in our formal mentoring program. His area Is why I need the certifications and 35 other people at the company have failed the test. I'm excellent at standardized tests. Passed all four parts of CPA exam on the first go and got a 790/800 on my GMAT and I'm studying for the test. Studying is how I passed the others. Is mentoring enough? Should I do more?

I don't think there is anything I can do about the manager who feels passed over. The VP feels he wasn't qualified for the job and she probably should not have told me he discussed it with her. I think she's concerned that he will make it challenging to get work done with his team.

Any other suggestions for handling the unexpected and very fast promotion? I haven't even fully figured out my old job.