I am now reading Bostrom and Sandberg's roadmap of brain emulation and it is a little bit old but still very good. It answers this question in details.
My impression is that current thinking is that connectome is enough. Other things I can think of the could matter:
But I think all of these are expected not to matter much because, for example, humans seem to come back the same as they were before after short periods of brain inactivity aside from damage from oxygen deprivation. Similarly neurotransmitter levels change the "mood" or "flavor" of a person, along with other chemicals that interact with neurotransmitters, but similarly these seem to be transient effects that change a person but are also easily tunable and so not necessarily a key part of what it means to be a particular person.
I think there is some case that interactions with parts of the body outside the brain might matter. For example, I find meditating is difficult when I am sick and have changes to my breathing. This suggests that something about this is interfering with the normal process that lets me get into a meditative state. But arguably this could easily be simulated and is not differentiating between humans, so may not and probably doesn't matter.
Thus to the best of my knowledge connectome is probably "enough" although more wouldn't hurt.
Epigenetic information might be important too (especially since epigenetic information may determine a neuron's sensitivity to "updates"). In metamorphosis, a butterfly seems to retain some of its caterpillar memories even though all of its neuronal arbors are completely reorganized and lost, and that information may be stored in the epigenome.
Also, there's a hypothesis that memory may even be stored in RNA (see Gaurav)
One question for WBE is whether we really need to upload a specific brain or if it is enough to use a generic brain as a template and a kind of informative prior to greatly speed up alternative AI techniques (an emerging paradigm I'm calling "brain imitation learning" for now).
There's a chapter in Rethinking Consciousness (unrelated to the rest of the book) about whole brain emulation, and I remember he definitely discusses the spatial resolution required on imaging each individual synapse in order to capture its type and strength.
Knowing a brain's connectome tells you which neurons can pass signals to other neurons, but presumably neurons differ in how they process that information. What are the most important things to know about an individual neuron in order to predict how it would fire given certain input signals? And to what extent is that information preserved when a brain is cryonically stored?