Synapses can be seen and analyzed via electron microscopy at 2-4 days postmortem in at least some cases
Many people will say that the brain decomposes completely within minutes after death. However, they usually don't offer data when they make such claims.
My impression from reading the literature is that actual postmortem decomposition is slower than many people think.
Here's an example from a recent article I read, Henstridge and colleagues 2015. They studied a brain tissue that they banked in part via immersion fixation in 4 % paraformaldehyde and 2.5 % glutaraldehyde in 0.1 M PB for 48 hours.
From Table 1, here is the pathoclinical information, including the postmortem interval. As you can see, some of the postmortem intervals prior to preservation are >3 days. It's unclear to me if the body/brain was refrigerated during this postmortem interval, but it's a likely possibility.
As a side note, this is a fascinating data set that includes intelligence test scores at age 11. This allows the researchers to adjust for premorbid cognitive functioning in a robust way as they investigate the causes of age-related cognitive decline.
After preparing the tissue for electron microscopy, they found that they were able to study synapses. Only a small percentage of the identified presynaptic and postsynaptic terminals were found to have degenerating profiles. This seems to have been attributed to antemortem Alzheimer's disease, rather than postmortem decomposition.
There are all sorts of confounds here. The study doesn't seem to have been focused on identifying the limits of the postmortem interval in which this sort of study is still informative. They may have adjusted for the postmortem interval and presence of decomposed synapses in some sort of way. Et cetera.
But generally speaking, this data tells us that it's reasonable to think that synapses might still be largely structurally intact even at 2-4 days postmortem interval. At least in some cases depending on the cause and circumstances of death. This is actual data, rather than pure speculation.