pretty much nothing would happen until these ships sail into each other's fire control radar horizons. Prior to that, assuming the Band Stand on the Lanzhou could target the Lassen, it would fire all 8 of the YJ-62's at the Lassen. All of these would be easily shot down, however.At the point that both these ships can actually target each other, which would be ~25km distance or less, the Lassen could immediately open fire with its Mk 45 5-inch gun, while the Lanzhou's main gun could not. It would have to close to within 17km distance; during this time it would have to endure punishing fire from the Lassen's main gun. Even worse, both the SM-2 and ESSM have anti-surface modes and these would also be launched against the Lanzhou as soon as it broke the radar horizon. By the time the Lanzhou closes to within 17km it would already have become seriously damaged or even sunk. It is unknown whether the HHQ-9 has an anti-surface capability, but even if it does, the larger magazine capacity of the Lassen means that it can outshoot the Lanzhou and still have extra missiles left for antishipping purposes. It is also possible that the SM-2 will be able to perform OTH self-targeting (Block IIIB) with its infrared seeker, in which case all of the Lassen's SM-2's could be launched against the Lanzhou long before it broke radar horizon, as long as the Lassen has a general idea of where the Lanzhou is.
Also, if the Lanzhou is loaded with rocket-boosted munitions, both ships could probably engage each other starting at ~25km range, though the Lanzhou would still be seriously disadvantaged by the fact that the Lassen could continuously riddle the Lanzhou with SM-2's and ESSM's, which individually don't do massive damage but could certainly destroy a ship if you pepper the ship with a large number of them, especially since they are mostly still loaded with fuel after such a short flight time. It is also possible that 25km is less than the minimum range of the SM-2, which I suspect depends on whether it has a booster (e.g. SM-2 Block IV, SM-6), in which case the Lassen would only have ESSM's available for antishipping. I do not have information on how many ESSM's are typically carried by a USN destroyer, but I suspect it's somewhere in the neighborhood of 16 (4 cells), or even 32 (8 cells).
Basically, there are a lot of unknowns in this scenario. Can the SM-2 be OTH launched against the Lanzhou? Maybe. Does the PLAN have rocket-boosted munitions yet? Maybe. Does the HHQ-9 have antishipping capability? Maybe. Is 25km too close for the Lassen to launch its SM-2's against the Lanzhou? Maybe. How many ESSM's and SM-2's does the Lassen carry? Unknown.
In real life the Lassen would never be caught alone if the chance of actual war breaking out was a real possibility. It would sail into the SCS as part of a CVBG with fighter CAP and AWACS flying overhead. The Lanzhou would be detected while still several hundred km away and get sunk by a squadron of F/A-18's loaded with several dozen Harpoons. Then again if the chance of war breaking out was a real possibility, the Lanzhou would also never be caught out by itself in open ocean.
