The European APAR X-band radar used on several air defense frigates already provide matching radar coverage for SM-2. And here we are talking about missiles like the ESSM or Type 3 which have officially ranges of 'just' >50km. A modern sole X-band radar system is more than good enough for self-defense or even point-defense while also providing accurate fire solution for its large main gun. I would feel safer on the Mogami than on the British Type 26 with its older low end Artisan radar system against aerial threats.
Artisan is a bit of a different animal. It is an E/F band radar, or shall we say, S-band. This band is favored for medium to long range search and is a bit more partially resistant to stealth. As the band attenuates less energy as its waves travel through the atmosphere, it will travel further. This is the reason why S-band is very common for volume search radars, both land and sea. Example: SPY-1.
APAR can reach 150km even with older tech. However, we don't know what RCS context is this, and brochures often don't mention and gloss over context so they can sound nicer. While you might be able to engage larger RCS targets at 150km, the same cannot be said of stealthier objects. Keep this in mind as engagement against cruise and antiship missiles tend to be much much closer than say a bomber. Context matters. If a radar advertises 300km while another radar advertises 250km, it may seem the first one is better. But if the context says 300km against bomber sized targets (RCS from 10m2 to 100m2) while the latter adverises against small fighter sized targets (2m2 RCS), the latter is better.
So why choose X-band? X-band is resistant to clutter, especially sea clutter. It is very good for scanning targets at the water surface or flying very low near the water surface. Its this reason why navigation radars use it. So if a small antiship missile is skimming very low near the water surface, an X-band radar can differentiate it and track it with less clutter and greater angular accuracy and resolution. Its for this reason why fire control radars use this band. This is good for shorter range defense. Even if BAE Systems would say their S-band radars can do the job, I would rather put my faith on X-band radars on closer range. Even if Artisan is able to put targeting data on a surface target, against a fast moving evasive flying target near the water surface I would put my bets on an X-band fire control radar running my dual purpose gunnery trying to nail the target down. In this context the Mogami might be more intended to work in shallower waters where clutter is more abundant.
Yet there is still another context. The RN Type 26 only has the Artisan, along with two navigation radars. The Mogami only has its X-band, with a navigation radar. But European frigates using the APAR like the Sachsen class, not only have the APAR, they also have the SMART-L. They have an X-band for the fire control and for dealing with sea skimmers, but they also have the longer band for volume search. They have the best of both worlds. SMART-L would reach 400km (against patrol aircraft; note once again context is important, this is down to 65km against stealthy missiles). Sachsen class plans to refit SMART-L with a new S-band radar, maybe Thales NS200, but that's another story. In any case, fitting two phase arrays sets with different bands is more expensive than one, so it should be expected that cost decisions will come in to the RN's City class as well as the Mogami class, frigates are supposed to be more price mindful otherwise they won't be called frigates.