We consider approximation algorithms for covering integer programs of the form min $\langle c, x \rangle $ over $x \in \mathbb{N}^n $ subject to $A x \geq b $ and $x \leq d$; where $A \in \mathbb{R}_{\geq 0}^{m \times n}$, $b \in \mathbb{R}_{\geq 0}^m$, and $c, d \in \mathbb{R}_{\geq 0}^n$ all have nonnegative entries. We refer to this problem as $\operatorname{CIP}$, and the special case without the multiplicity constraints $x \le d$ as $\operatorname{CIP}_{\infty}$. These problems generalize the well-studied Set Cover problem. We make two algorithmic contributions. First, we show that a simple algorithm based on randomized rounding with alteration improves or matches the best known approximation algorithms for $\operatorname{CIP}$ and $\operatorname{CIP}_{\infty}$ in a wide range of parameter settings, and these bounds are essentially optimal. As a byproduct of the simplicity of the alteration algorithm and analysis, we can derandomize the algorithm without any loss in the approximation guarantee or efficiency. Previous work by Chen, Harris and Srinivasan [12] which obtained near-tight bounds is based on a resampling-based randomized algorithm whose analysis is complex. Non-trivial approximation algorithms for $\operatorname{CIP}$ are based on solving the natural LP relaxation strengthened with knapsack cover (KC) inequalities [5,24,12]. Our second contribution is a fast (essentially near-linear time) approximation scheme for solving the strengthened LP with a factor of $n$ speed up over the previous best running time [5]. Together, our contributions lead to near-optimal (deterministic) approximation bounds with near-linear running times for $\operatorname{CIP}$ and $\operatorname{CIP}_{\infty}$.