As @Zhe points out, its not possibly to definitively answer your questions without knowing the structure of the olefin, as its important what is around the olefin, as well as just how many protons are attacked to the olefin.
If you read many of the original Grubbs' papers (and indeed any papers using a Grubbs' metathesis to make tri-substituted alkenes), the common method to determine the geometry is to use the nuclear Overhauser effect, or nOe for short.
The nOe identifies interactions through space (as opposed to standard NMR where we're looking at the interactions through bonds. By recording the nOe, one can essentially 'measure' how close two protons are, which in this case would allow you to distinguish E and Z.
Practically speaking, there are several different NMR experiments that one can run to observe the nOe.
- The 1D nOe (1D-NOESY) difference experiment involves irradiation of a particular signal (for instance the one alkene proton), to allow measurement of what other protons are proximal in space. This experiment has the advantage of being quick to run, but in order to gain useful information, you have to run several of them (irradiate the alkene proton, then run a different 1D-NOESY in order to make sure that the proton that the alkene proton is 'seeing' also 'sees' the alkene proton
- The 2D nOe (2D-NOESY) experiment shows all nOe enhancements in a molecule, this is incredibly useful and often used in structure elucidation, but can be slow to run, especially if the amount of sample available is limited.