Consider an even positive integer $n$. The task is to determine the number of such $n < 3^{2022}$ that can be partitioned into $\frac{n}{2}$ two-element subsets such that the sum of the elements in each subset is a power of $3$. Such an integer $n$ is termed as "nice".

### Step 1: Pairing Elements

We begin by examining the set $\{1, 2, \dots, n\}$. We need to partition this set into pairs $\{a, b\}$ such that:
$$a + b = 3^k$$
for some integer $k$.

### Step 2: Power of $3$ Structure Analysis

The powers of $3$ can be expressed as:
$$3^0 = 1, \, 3^1 = 3, \, 3^2 = 9, \, \ldots$$

Given that $n$ is even, the sum of all numbers from $1$ to $n$ is:
$$S = \frac{n(n+1)}{2}$$

### Step 3: Nice Condition

For $n$ to be nice, the total sum $S$ should permit pairing into sums that are powers of 3. Knowing:
$$a + b = 3^k$$
and since $a, b \in \{1, 2, \dots, n\}$, we must ensure that for each power of $3^k$, the elements $a$ and $b$ exist within the set.

### Step 4: Determining Number of "Nice" Integers

We are tasked with ensuring that $n$ can match up to various powers of 3. The key lies in understanding the binary representation that enables such a partition.

For an integer $n$ to be partitioned in such a way, $n$ must take the form:
$$n = (3^1 - 1) \cdot 2^0 + (3^2 - 1) \cdot 2^1 + \cdots + (3^{\lambda} - 1) \cdot 2^{(\lambda-1)}$$
where each term implies the possibility of partitioning into $\left(3^k - 1\right)/2$ pairs for different $k$.

### Step 5: Counting

To find the number of "nice" integers smaller than $3^{2022}$, observe that each power contributes a degree of freedom through its binary representation. Specifically, the possibility arises if:

An even $n$ can be expressed as some configuration of formed sums listed above where each $3^k$ contributes as a viable pairing option.

Now, every unique combination of existence or absence (binary relevant state) of a situational $2^k$ corresponds to an allowed $n$. For $2022$ such configurations:

There are $2^{2022} - 1$ valid combinations (as all must select at least one viable condition for pairing).

Thus, the number of nice integers smaller than $3^{2022}$ is:
$$\boxed{2^{2022} - 1}$$