Anti sub helos and MPAs are not the most difficult targets... but then imagine an arsenal sub based on the Akula SSBN with UKSK-M launch tubes... you could fit the sail with enormous AESA radar panels and IR sensors and have it sit on the surface launching thousands of SAMs to defend a group of ships... a SAM and radar picket frigate would have nothing on it....
It's a good idea, but not in the way of an anti-air asset for a naval group.
Surface combatants have their own anti-air radars and missiles, and there will typically be a ship with a Redut system capable of launching missiles at up to 120km ranges. It's sufficient. Their radars will be further above the horizon than the sub's would be. Surface vessels are a lot more survivable than subs; one missile getting through won't take them out. They are also immediately ready for action at any moment without having to surface to track and engage targets, or even be aware that there are targets. There is no real advantage to having a surface group rely on a sub as its most capable anti-air asset. You might as well have a dedicated large surface ship for that if that's what you want, or better yet a multirole vessel - both would be a lot cheaper than an Akula and more capable.
An anti-air Akula's real advantage lies in its ability to surface in any unexpected place at any time and turn an area of assumed safe airspace into a denied one. It can catch entire flights, or vulnerable planes such as transports, AWACS, reconnaisance, etc... completely off-guard and destroy them before the hostile airforce even realizes where the missiles are coming from. After the sub has destroyed a few targets, it can submerge, retire and then surface somewhere else at another time; this will save it from anti-radiation missiles. Anti-air guerilla warfare if you will. The danger of course is the enemy promptly deploying ASuW aircraft and warships to hunt for it; they will know its last known position to a high degree of accuracy and can conduct sweeps from there, deploy sonar buoys along presumed routes, deploy autonomous torpedoes to hunt for it and so on. An under-water anti-air launch capability for SHORAD class missiles and some sort of deployable floating short-range radar array might be neccessary for self-defence from ASuW aircraft, together with extensive decoy and anti-torpedo measures.
All in all not cheap, especially for a one-off vessel. An Akula could be used as a test-bed to experiment with the concept, with various ad-hoc temporary additions and equipment - but a smaller, serial sub design modified from a more modern model already in production might be optimal for actually implementing it; with a deployable radar mast with IR sensors and so on.