Is there a simple way to calculate/estimate the volume of Minkowski sum of an ndimensional unit ball and an ndimensional ellipsoid? Even a simple ellipsoid like $\frac{x_1^2}{a^2} + x_2^2 + \ldots + x_n^2 = 1$ will do.

I guess you know that the result can be written as a polynomial. $$w_0+w_1{\cdot}r+\dots+w_n{\cdot}r^n.$$ So your question is how to estmate the coefficients $w_i$; this is so called "crosssectional measures" and they can be defined for any convex body $K$.
If you want to write $w_i$ as an integral, check for example BuragoZalgaller, Geometric inequalities. For example $$w_i=\mathrm{MayBeAConst}\cdot\int\limits_{\partial K} \sigma_{i1}(k_1,k_2,\dots,k_{n1})\\, d\mathrm{area}.$$ where $\sigma_i$ is the $i$th elementary symmetric polynomial and $k_i$ are principle curvatures. In your case it is easy to find $k_i$... 


Because your ellipsoid has all axes but one of equal length we don't need the "full theory": Let $x_1=:x$ and $(x_2, \ldots, x_n)=:y\in{\mathbb R}^{n1}$. The Minkowski sum $M$ of the unit ball and the given ellipsoid is rotationally symmetric with respect to the $x$axis and consists of all points $z=(x'+x'', y'+y'') \in {\mathbb R}^n$ with $${x'^2\over a^2}+y'^2\leq 1\ ,\qquad x''^2 +y''^2\leq 1\ .$$ A hyperplane $x={\rm const.}$ intersects $M$ in an $(n1)$dimensional ball whose radius can be found by maximizing $$y'+y''\leq y'+y''\leq \sqrt{1x'^2/a^2} + \sqrt{1x''^2}$$ under the constraint $x'+x''=x$. 


For more on this, check out "Surface area and other measures of ellipsoids", by one I. Rivin: [math/0403375] Surface area and other measures of ellipsoids by I Rivin ► Surface area and other measures of ellipsoids  Elsevier by I Rivin  2007 Advances in Applied Mathematics 39 (2007) 

