What you ask does not have a simple answer and the ability to calculate palaeodepths will vary depending on what data you have available. It will also become more difficult depending on the location and how far you go back in time, as tectonic factors will complicate matters.
There will also most likely be a significant degree of uncertainty.
Broadly speaking though, the factors you will need to consider will be:
Eustasy - ice-volumes for the period you are looking at either leading to a reduction or increase in depth relative to contemporary.
Isostasy - Is your location affected by the presence/absence of ice? Has it undergone uplift of subsidence. Is it in a fore-bulge region?
Tectonics - is the location you are studying tectonically active? Has it been for a period that influences the period you are looking at? Over longer time periods this influences basin geometry and therefore volume.
Dynamic topography - Mantle plumes lead to significant shifting of the position of the crust, as has recently been demonstrated looking on the eastern US margin (see Rowley et al. 2013, Science vol 340, p. 1560-1563).
Subsidence via increased sediment deposition. For example a sediment fan will lead to crustal depression.
Sedimentation - you may have to back-strip sediments deposited in the intervening period. This in turn will influence subsidence/uplift.
I don't mean to over complicate things - but there is a lot going on! So the further back in time you are looking the harder it is. Considering the tropical Atlantic 20 ka ago it is a fairly simple matter, looking in the eastern Mediterranean 40 Ma ago will be harder (at this point I think it was still the Tethys).
In my opinion one of the better methods to do this involves looking at the palaeoshorelines, for instance the Orangeburg Scarp, which demarcks a Pliocene shoreline in the US. It is also possible (on a coarse scale) to look at benthic foraminiferal tests and see if you have any species that are depth limited.
Again though, these measures are influenced by those factors I listed above.
I think the best advice I could give would be to try and find as much information concerning the area and period you are looking at, and once you know what data you have available then make a decision about whether it is possible to reconstruct the palaeodepth.
If you do, then you can begin to look at the specific influences in the location and the begin to try and assign numbers. Though again, depending on where you are there may be large uncertainty.
I dont know how old your sediments are or where they are from but one way is to look at the ratio between planktonic and benthic foraminifera. Although, depending on your setting there may be complications and hence this method may not be suitable. Benthic foraminiferal assemblages can also be used to help assess relative depths.
see for example:
Van der Zwaan, G. J., Jorissen, F. J., and de Stitger, H. C., 1990. The depth dependence of planktonic/benthic foraminiferal ratios: Constraints and applications. Marine Geology, 95: 1-16.
Benthic foraminifers are ubiqutous and their distributions are mainly affected by several chemical and physical factors. One of them is the water depth. In the Deep Sea Drilling Program (DSDP), or IODP and ODP benthic forams are being used to possible age-date the samples brought from the drilling hole and also to determine the paleodepth which means the depth that the species used to inhabit at. The depth here is relatively determined based on categories (Shallow, Abyssal.. etc). So, if you are looking for an estimate of the depth in some location, you may want to use a specialist in benthic foraminifera. I hope this helps.