Consider any numerical integration method $I^*(f,a,b)$ that approximates the exact integral $I$ of a function $f$ over an interval $[a,b]$. It will be implemented by a computation represented by, say, `I[f[x], {x, a, b}]`. Conceiving an `NIntegrate[]` command in this way breaks the error into two independent components. The first is the error of the method $I^* - I$, where $I^* = I^*(f,a,b)$ is assumed to be computed exactly; this error is often called the "truncation error." The second is the difference between $I^*$ and `I[f[x], {x, a, b}]`; this error is usually thought of as "rounding error."

The error estimator of an `NIntegrate[]` method endeavors to yield an ***upper bound*** on the truncation error.  The options [`PrecisionGoal`](http://reference.wolfram.com/language/ref/PrecisionGoal.html) and [`AccuracyGoal`](http://reference.wolfram.com/language/ref/AccuracyGoal.html) set the goal for this bound. The error estimate is often done by comparing the method with a less accurate method, e.g. in the [`"GaussKronrodRule"`](http://reference.wolfram.com/language/tutorial/NIntegrateIntegrationRules.html#381359375) and the [`"ClenshawCurtisRule"`](http://reference.wolfram.com/language/tutorial/NIntegrateIntegrationRules.html#486402291).  For a well-behaved function, with one of these two rules, the error can be greatly overestimated, as noted in [this answer](http://mathematica.stackexchange.com/a/119390/4999).

The option [`WorkingPrecision`](http://reference.wolfram.com/language/ref/WorkingPrecision.html), which controls the [`Precision[]`](http://reference.wolfram.com/language/ref/Precision.html) and [`Accuracy[]`](http://reference.wolfram.com/language/ref/Accuracy.html) of the result, is used to control rounding error.  (It also affects the default `PrecisionGoal`, which affects the bound on the truncation error discussed above and which [@ilian](http://mathematica.stackexchange.com/users/145/ilian) also notes in [his answer](http://mathematica.stackexchange.com/a/126645/4999).)  `Precision@arb16` and `Accuracy@arb16` reflect this rounding error.  Normally this, too, is an upper bound.  So `Precision[]` has little to do with the truncation error in approximating the integral.

The general approach in numerical analysis has tended to focus on bounding error.  *Mathematica's* error estimators and arbitrary-precision numerics follow this approach. I don't believe there is a way to get a better estimate of the error, without knowing the exact value of the integral or computing it to a higher precision. (One could perhaps use a rule with a good error estimator; the approach in the rule `chebRule` in the linked answer above does somewhat better on a function that is analytic on `{0, 1}` than `"GaussKronrodRule"`, but the OP's function has a singularity at `1`.)