When will acid-base reactions occur before substitution reaction For example, given the reaction below:



The substrate is secondary, the solvent is aprotic and it is to react with a strong bulky base. From this, I assumed that the reaction would be $\ce{E2}$. However, it instead begins with an acid-base reaction where the tert-butoxide removes the proton of hydroxide and then proceeds with a $\ce{Sn1}$ reaction where the oxygen attacks the carbon of the leaving group to form an epoxide. 

How can you determine if an acid-base reaction will take place before proceeding with a substitution reaction?
Acid-base reactions (like the one shown, involving proton transfer between electonegative atoms) are almost always extremely fast - for all practical purposes instantaneous (diffusion-controlled), and therefore much faster than substitution reactions. In the example given, the bromocyclohexanol is of comparable acidity to tert-butanol (maybe even more acidic), so we would expect instant proton transfer, and the observed result.

The initial proton transfer might not be so favorable if you used a much weaker base, so you could get a different result - or no reaction.