Realistically,what you need to get is the support of at least one of the permanent members of the security council. To actually succeed, you'd want the US, Russia, China supporting you and France being not hostile. The UK will follow the lead of the US in Middle East issues and can be ignored. Under these circumstances you can let your friends handle the UN and ignore it.
Your question is somewhat puzzling as it seems to suggest that the UN has some independent political clout, which it specifically does not. Preventing such influence is why those permanent members have their veto power.
As for implementing this... that is left as an exercise to the reader. Just kidding. The best method is generally to take it step by step and opportunistically. Have a clear idea what you want to do and act decisively when an opportunity arises. That is how Bismarck united Germany and defeated France and Austria. (And Denmark.)
First, he inherited a state capable of acting more efficiently than the competition and built up on that. So you must build a military that is capable of gaining your goals with a short, victorious war. The important thing here that Bismark understood, but his successors did not, is that instead of trying to build an insane war machine you break up your goals into to smaller steps that the military you actually have can achieve and then do those step by step. This "lack of ambition" also stops other players from going to panic mode and ganging up on you.
And that really is the most important part, use diplomacy to stop others from joining forces against you. Bismarck had wars against most of his significant neighbours while having significant internal instability due to unification. He was able to to defeat each of his enemies in isolation, often with the help or support of his next target. Or the previous one.
Doing this requires using "real politik". Your actions must be based on the actual reality on the ground and rational decision making. Not on some obscure historical enmity between your country and, say, Israel or Iran. Your alliances must be issue specific and you must ally with every one with significant power that supports you on the issue.
Then you simply must choose such issues that you can further your ultimate goal of uniting the Middle East at the expense of one player who gets isolated in the issue. Most politicians have problems handling the issues that arise due to external factors. True players of the "real politik" know that you are entirely free to interpret those issues as you wish.
If Austria throws a fit as a consequence of an assassination and insists the existing alliance requires Germany to support them in the resulting war, the German Chancellor is entirely free to disagree and instead insist on a peace conference with an agenda of his making. Even if you have the strongest army in the area, you must always remember that it is impossible to win a war. Victory comes at the negotiating table. Fighting just gives you an edge you can use. Thus you should move to peace as fast as is practical. Preferably before a single shot has been fired at your forces. Foreign Archdukes do not count.
Minimizing warfare while at a position of strength also makes diplomacy much easier. This also covers all hostility. If you have hostile relations with someone, either inherited or because you just had a war with them, you should move to have friendly relations with them as soon as you can. Potential allies are useful, potential enemies are not. Any wars you have must be to solve a specific issue, not to fight an enemy.
You are probably wondering why I am listing general remarks, not specific notes on Middle East. That is because the situation evolves from the current as soon as you take the first step. In fact the major goal of your first steps, any steps really, should be to transform the situation so that the reality and "the gaming board" changes. Not necessarily directly in your benefit, but enough that alliances and old enmities become fluid.
Note that the result is a tangle of alliances and pacts that only a really good player can manage. That would be you, and optimally not anyone you have a conflict of interest with. So you should have a plan of succession that either gives your successor an easier to manage situation or assure a successor capable of replacing you.
For Egypt, I'd suggest the first step should be pacification of Libya. Libya has oil. Egypt has strong interests in Libya that allow you to raise an issue in Libya any time you wish. And nobody really cares, what you do to the lunatics currently ruling much of the country. Especially not the ordinary Libyans, they must be fairly disillusioned with the way things currently are.
And it is fairly easy to raise a counter-terrorism angle to get support from the US. China can be bough with oil. The Europeans with oil and fixing the humanitarian situation. (Read: African immigration issue.) Russians could be swayed with a more rational and independent Middle East policy. They do not really approve how Uncle Sam and friends have been mucking around. Note that you should use deals with factions in Libya as much as you can and minimize fighting. Since your plans are bigger than Libya you can afford to be generous and give wide autonomy in addition to peace and stability.
Usually there would be a nationalism issue with Libyans not really liking being annexed by Egypt, but since you are intending to rebuild a caliphate anyway, you can just use that to gain popular support. And the need for popular support to explain why you are talking about caliphate. Which totally would screw over those people in Syria that everyone except possibly Turkey hates... (Because they fight Kurds.)
If you can pacify and annex Libya, next step should probably be Sudan. Using negotiation annex southern parts as autonomous area and then crush the northern part with its religious lunatics in between. Rationales and external politics should be similar to Libya.
Then you could negotiate peaceful annexation with what remains of the Assad regime and move to pacify and annex Syria.
For the rest of region, there would be too many changes at this point to make reasonable predictions, but if you did the negotiation parts of the previous annexations well enough, you should have wide popular support for your caliphate and lots of credibility with the political elite. Opportunities should arise naturally. You should be able to offer people of Iraq a better deal than their own government, for example.