How much climate history can be reconstructed using marine sediment cores?
The oldest in situ marine sediments recovered by IODP (International Ocean Drilling Program; formerly ODP and DSDP) was (from memory) Late Triassic (so ca. 200 to 230 Ma) and it is as old as in situ marine sediments go, since indeed, as the marine crust gets older, it gets denser and thus is more prone to be subducted. However we do have older marine sediments, that were lifted during various orogenies and that were somewhat metamorphosed but are still viable for paleoclimatic studies (to some reduced extent of course). These lifted and metamorphosed sediments (along with the crust below them and even the upper mantle dragged along) are referred to as ophiolites, and they indeed date as far back as the earliest Phanerozoic (so 500 Ma). As most of the climatic information we are getting from them depends on the planktonic or benthic eukaryotic activity at the time of sedimentation, we cannot really go further back with these.