Suppose that the statement is false and $f(\mathbb{R})=\mathbb{N}$. We prove several properties of the function $f$ in order to reach a contradiction.

To start with, observe that one can assume $f(0)=1$. Indeed, let $a \in \mathbb{R}$ be such that $f(a)=1$, and consider the function $g(x)=f(x+a)$. By substituting $x+a$ and $y+a$ for $x$ and $y$ in (1), we have
$$g\left(x+\frac{1}{g(y)}\right)=f\left(x+a+\frac{1}{f(y+a)}\right)=f\left(y+a+\frac{1}{f(x+a)}\right)=g\left(y+\frac{1}{g(x)}\right) .$$
So $g$ satisfies the functional equation (1), with the additional property $g(0)=1$. Also, $g$ and $f$ have the same set of values: $g(\mathbb{R})=f(\mathbb{R})=\mathbb{N}$. Henceforth we assume $f(0)=1$.

Claim 1. For an arbitrary fixed $c \in \mathbb{R}$ we have $\left\{f\left(c+\frac{1}{n}\right): n \in \mathbb{N}\right\}=\mathbb{N}$.

Proof. Equation (1) and $f(\mathbb{R})=\mathbb{N}$ imply
$f(\mathbb{R})=\left\{f\left(x+\frac{1}{f(c)}\right): x \in \mathbb{R}\right\}=\left\{f\left(c+\frac{1}{f(x)}\right): x \in \mathbb{R}\right\} \subset\left\{f\left(c+\frac{1}{n}\right): n \in \mathbb{N}\right\} \subset f(\mathbb{R})$.
The claim follows.

We will use Claim 1 in the special cases $c=0$ and $c=1 / 3$ :
$$\left\{f\left(\frac{1}{n}\right): n \in \mathbb{N}\right\}=\left\{f\left(\frac{1}{3}+\frac{1}{n}\right): n \in \mathbb{N}\right\}=\mathbb{N} .$$

Claim 2. If $f(u)=f(v)$ for some $u, v \in \mathbb{R}$ then $f(u+q)=f(v+q)$ for all nonnegative rational $q$. Furthermore, if $f(q)=1$ for some nonnegative rational $q$ then $f(k q)=1$ for all $k \in \mathbb{N}$.

Proof. For all $x \in \mathbb{R}$ we have by (1)
$$f\left(u+\frac{1}{f(x)}\right)=f\left(x+\frac{1}{f(u)}\right)=f\left(x+\frac{1}{f(v)}\right)=f\left(v+\frac{1}{f(x)}\right)$$
Since $f(x)$ attains all positive integer values, this yields $f(u+1 / n)=f(v+1 / n)$ for all $n \in \mathbb{N}$. Let $q=k / n$ be a positive rational number. Then $k$ repetitions of the last step yield
$$f(u+q)=f\left(u+\frac{k}{n}\right)=f\left(v+\frac{k}{n}\right)=f(v+q)$$
Now let $f(q)=1$ for some nonnegative rational $q$, and let $k \in \mathbb{N}$. As $f(0)=1$, the previous conclusion yields successively $f(q)=f(2 q), f(2 q)=f(3 q), \ldots, f((k-1) q)=f(k q)$, as needed.

Claim 3. The equality $f(q)=f(q+1)$ holds for all nonnegative rational $q$.

Proof. Let $m$ be a positive integer such that $f(1 / m)=1$. Such an $m$ exists by (2). Applying the second statement of Claim 2 with $q=1 / m$ and $k=m$ yields $f(1)=1$.
Given that $f(0)=f(1)=1$, the first statement of Claim 2 implies $f(q)=f(q+1)$ for all nonnegative rational $q$.

Claim 4. The equality $f\left(\frac{1}{n}\right)=n$ holds for every $n \in \mathbb{N}$.

Proof. For a nonnegative rational $q$ we set $x=q, y=0$ in (1) and use Claim 3 to obtain
$$f\left(\frac{1}{f(q)}\right)=f\left(q+\frac{1}{f(0)}\right)=f(q+1)=f(q) .$$
By (2), for each $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f(1 / k)=n$. Applying the last equation with $q=1 / k$, we have
$$n=f\left(\frac{1}{k}\right)=f\left(\frac{1}{f(1 / k)}\right)=f\left(\frac{1}{n}\right) .$$
Now we are ready to obtain a contradiction. Let $n \in \mathbb{N}$ be such that $f(1 / 3+1 / n)=1$. Such an $n$ exists by (2). Let $1 / 3+1 / n=s / t$, where $s, t \in \mathbb{N}$ are coprime. Observe that $t>1$ as $1 / 3+1 / n$ is not an integer. Choose $k, l \in \mathbb{N}$ so that that $k s-l t=1$.
Because $f(0)=f(s / t)=1$, Claim 2 implies $f(k s / t)=1$. Now $f(k s / t)=f(1 / t+l)$; on the other hand $f(1 / t+l)=f(1 / t)$ by $l$ successive applications of Claim 3. Finally, $f(1 / t)=t$ by Claim 4, leading to the impossible $t=1$. The solution is complete.