Methanol vs Ethanol Substrate cleaning
Both $\ce{SrTiO3}$ and $\ce{MgO}$ can safely be washed with methanol. These compounds are insoluble in most any solvent other than certain acids. I suspect that any online reference you've found saying methanol is corrosive is with respect to organic compounds, nothing like the composition of your substrates. 

One (somewhat handwaving) way to convince yourself that this is the case is that we know the compounds are insoluble in water (see their Wikipedia and PubChem pages) and ethanol (from your experience, also confirmed for $\ce{MgO}$ in it's Wikipedia and PubChem pages). In terms of solvent polarity, methanol lies between water and ethanol. It would be extremely unlikely for an ionic solid to be completely insoluble in water and ethanol yet have some significant solubility in methanol.  

Although we can be confident that the substrate materials will not be destroyed by methanol, only trial will let you know how effective it is for cleaning whatever contamination may be present in your situation. Methanol is fairly, though not completely, similar to ethanol in it's solvent properties, and there is a good chance that it will be as effective as ethanol. Again though, only trial will verify this in your situation.