This is pretty broad question so I'm not really sure what the end goal is here, but here goes.
The answer is both: "yes" and "it depends what you mean exactly". I'll explain what I mean.
Firstly, you should clarify in your question whether what you actually mean is the substitution rate, or not. The mutation rate, which is how often a given base pair position changes, cannot be observed with any current technique, even Next Generation Sequencing. This is for 2 reasons (that spring to mind, there may be more).
* If you make an observation at time $t$, and another at time $t +1$, but a base mutates (say A $\rightarrow$ C) and then mutates back again between your observations, you would not be able to tell that this had ever happened. Thus there would be 2 mutation events which you would not count in your 'rate'. 882
This is one of the reasons molecular clocks based on SNPs tend to underestimate ancestral distances.
* If a mutation occurs that is deleterious (has a negative effect on the organism), the mutation won't be observed because the lineage of the organism won't survive to be sampled. By the way, the chances of a mutation being deleterious are considerably higher than it being advantageous, but lower than it being neutral.
The substitution rate is how many mutations have occurred that persist in the lineage. Now if we assume that when you say mutation rate, you mean substitution rate...
Yes, you can study mutation rate without 'molecular techniques'.
Throughout the history of classical genetics we've studied mutation rate without access to these tools.
My favourite example is basically the one I used in this question, where you can study the emergence of antibiotic resistance in bacteria (since you provided the example of E. coli, this is rather apt).
In the papers I linked however, they will almost certainly have used some molecular techniques to validate their answers.
But, if we set that aside for the moment, the very first indication of antibiotic resistance one might observe in the lab is seeing colonies of a bacteria growing on a petri dish in the presence of an antibiotic they are supposedly susceptible to.
Now, if we consider the 'mutation rate' to be the time taken (perhaps counted in the number of generations), for a spontaneous mutation to appear as a phenotype (observed characteristic), you can assess this purely visually without any molecular techniques. If you repeated this experiment however, you would most likely not be able to verify that resistance arose in exactly the same way twice without use of various molecular technqiues.
As a molecular biologist though, I would not really consider this a very good approach/definition of mutation rate, which leads me on to...
'It depends what you mean exactly'
If you were to stick to what is probably the stricter but more widely accepted definition of mutation rate (.i.e substitution rate), you'd need molecular techniques to determine this, as you'd need to know precisely which bases are changing.
Hand-wavy list of pros and cons:
Phenotyping (pros and cons):
* Quick
* Cheap
* Easy
* Low resolution
* Possibly not reproducible
Genotyping (pros and cons):
* Massive resolution
* Can be quick in some circumstances
* Functional information
* Generally expensive
* Requires specialist equipment
Not an exhaustive list by any means. 225