When you have a covering map $p:\tilde{X}\to X$ with a CW structure on $X$, you can use the homotopy lifting property to lift each cell of $X$ to a collection of cells of $\tilde{X}$, since you can think of maps $D^k\to X$ as being homotopies if you parameterize things right. In particular, given a cell $e^k:D^k\to X$, there is a map $\tilde{e}^k:D^k\to \tilde{X}$ called a lift such that $p\circ \tilde{e}^k=e^k$. Lifts are not unique.
Let's say $p$ is a universal covering space for simplicity.
The $0$-cells in $X$ each lift to a collection of $0$-cells in $\tilde{X}$ in correspondence to $\pi_1(X)$. It's easier to keep track of things when $X$ has a single $0$-cell that is the basepoint: then the $0$-cells of $\tilde{X}$ are in correspondence with $\pi_1(X,*)$. Since $0$-cells are just points, lifts are points from the inverse image $p^{-1}(*)$. Let's say $X$ has a single $0$-cell for simplicity.
The $1$-cells in $X$ are then loops (since the boundaries are attached to the single $0$-cell $*$), and so they can be thought of as elements of $\pi_1(X,*)$. These lift to paths between lifts of the basepoint. In particular, if a $1$-cell $e^1$ corresponds to $a\in\pi_1(X,*)$, and if $*_x\in\tilde{X}$ is a lift of the basepoint corresponding to $x\in\pi_1(X,*)$, then there is a lift of $e^1$ that is a path from $*_x$ to $*_{xa}$. This is essentially reiterating part of the construction of the universal covering space.
The $2$-cells and higher are simply connected, and so there is one lift of each per lift of the basepoint. It is a little tricky to figure out what happens to the attachment map, but usually you can see the order of the $1$-cells along the boundary, then follow the lifts of those cells in the cover.
In your case of $S^1$ with one $0$-cell and one $1$-cell, the lifts of the $0$-cell are $\mathbb{Z}\subset\mathbb{R}$. The loop lifts to paths from $n$ to $n+1$. So, the CW structure of $\mathbb{R}$ that you describe is the one coming from the universal covering map $t\mapsto e^{2\pi i t}$.
