Someone who thinks he's very important, but isn't

Solution 1:

There is no word-for-word translation, and no single word that aptly encompasses the idea, color, and register of the Russian original. Rather, you have to step a bit back and translate your sentence as a whole.

The English equivalent to "у него синдром вахтера" is

That's the tiniest/smallest amount of power I have ever seen go to anyone's head.

This closely matches the Russian idiom in every respect, and is a perfectly idiomatic retort in the situation that you've quoted.

Solution 2:

I am not aware of any English idiom that closely matches the Russian one you mention. Somewhat similar is the idiomatic set phrase paper tiger, which Merriam-Webster's Eleventh Collegiate Dictionary (2003) defines as follows:

paper tiger n (1850) : one that is outwardly powerful or dangerous but inwardly weak or ineffectual

On the topic of inflated self-regard, Wolfgang Mieder, A Dictionary of American Proverbs (1992) points to several English proverbs that seem on point:

Conceit is nature's gift to small men to make up for that which they don't have.

Conceit is God's gift to little men.

Every man has a right to be conceited until he is successful.

There is also an Aesop's fable that delivers a related moral. From "The Gnat and the Bull," in The Aesop for Children:

A Gnat flew over the meadow with much buzzing for so small a creature and settled on the tip of one of the horns of a Bull. After he had rested a short time, he made ready to fly away. But before he left he begged the Bull's pardon for having used his horn for a resting place.

"You must be very glad to have me go now," he said.

"It's all the same to me," replied the Bull. I did not even know you were there."

[Moral:] We are often of greater importance in our own eyes than in the eyes of our neighbor.

[Alternative moral:] The smaller the mind the greater the conceit.

Solution 3:

The nearest I can think of is delusions of grandeur:

the belief that you are more important or powerful than you really are

This is used as a term in psychology but also informally for someone who is a bit above themselves. The dictionary page gives a few synonyms that might be relevant (eg officious, smart-alec, tin god, throw your weight. around ...)

Close, also, is jobsworth:

someone who always obeys all the rules of their job even when they cause problems for other people or when the rules are silly

This one is, I think, British English (only?)

Solution 4:

I believe the closest analogue in English to someone having "синдром вахтёра" would be to say that they are power tripping, or having a power trip.

See power trip:

An activity or way of behaving that makes a person feel powerful: something that a person does for the pleasure of using power to control other people

From Oxford Languages:

a self-aggrandizing quest for ever-increasing control over others

There are some differences from Russian:

Someone who is power tripping might have actual power; the person abuses it to feel self important. In Russian, the implication is that the person is decidedly not powerful, which drives their desire to abuse whatever modicum of power that they have. In practical usage, the people who are said to be power tripping do fall into that category: forum moderators, policemen, security guards, rent-a-cops, low-tier managers, etc.
"Синдром Вахтёра" seems to describe a more permanent state of mind, as if it were a personality disorder; "power tripping" can refer to a specific action, or a period of time. A person (say, a policeman on duty) can go power tripping one day (abusing his power to self-aggrandize), but that doesn't imply that they are always like that. So someone has "Синдром Вахтёра" if they are power-tripping all the time while holding a position of very little actual power. Due to this, the effects of someone having "синдром вахтёра" on others are limited to annoyances. Someone on a power trip, on the other hand, can cause any amount of damage (e.g.: "power-tripping cops are killing innocent people") because there is no implication that the power they have is insignificant.
The expression power trip can refer to a specific act (e.g. a policeman ordering someone to speak politely, and jailing them for being impolite); "синдром вахтёра" doesn't refer to a specific act, but rather a persistent pattern of behavior. That said, a power-tripping person is pretty close to saying that it's a person with "синдром вахтёра".