Мы напрасно ждали автобус у другого входа и только потратили время впустую.

Because the verb ждать usually takes a direct object in Russian.

So:

• ждать автобус = to wait for the bus
• ждать друга = to wait for a friend
• ждать поезд = to wait for the train

English uses wait for, but Russian normally does not use a preposition here.

In this sentence, автобус is in the accusative case. Since автобус is masculine and inanimate, its accusative form looks the same as the nominative:

• nominative: автобус
• accusative: автобус

So even though the form does not change, it is still functioning as the object of ждали.

Because the preposition у normally requires the genitive case.

So:

• у входа = by the entrance / at the entrance
• у другого входа = by the other entrance

Both words change to genitive singular:

• другой → другого
• вход → входа

This is a very common pattern in Russian:

• у дома = by the house
• у станции = by the station
• у двери = by the door

Here у means something like by, near, or at.

So у другого входа means:

• by the other entrance
• near the other entrance
• in context, simply at the other entrance

It does not necessarily mean being directly inside or on something. It often suggests location next to or in the area of something.

Compare:

• у входа = by the entrance
• входе would relate to being in/on the entrance, which would not fit here

So у is the natural choice for saying where they were waiting.

Yes, they are very similar, and that is exactly why both are used here: for emphasis.

The sentence says:

• Мы напрасно ждали автобус...
• ...и только потратили время впустую.

A natural English sense is:

• We waited for the bus there for nothing and just wasted our time.

Using both words is not a mistake. Russian often allows this kind of repetition of similar ideas to make the statement stronger and more expressive.

So the sentence has the feeling of:

• We waited there uselessly, and all it did was waste our time.

They are close in meaning, but the nuance is slightly different.

• напрасно = for nothing, in vain, without result, sometimes also needlessly
• впустую = to no purpose, wastefully, for nothing, often with the idea that time, effort, or resources were wasted

In this sentence:

• напрасно ждали = waited in vain
• потратили время впустую = wasted time

So впустую works especially well with time, effort, energy, money, etc.

Examples:

• Я напрасно волновался. = I worried for nothing.
• Мы потратили деньги впустую. = We wasted the money.

This is a very common and important Russian pattern.

• ждали is imperfective
• потратили is perfective

Why?

ждали (imperfective)

This describes the process of waiting. The sentence focuses on the fact that they were engaged in that activity.

• Мы ждали автобус... = We were waiting for the bus...

потратили (perfective)

This describes the completed result: the time ended up being wasted.

• потратили время = spent/wasted the time

So the sentence combines:

• an ongoing activity: waiting
• a completed outcome: wasted the time

That contrast is very natural in Russian.

Because the subject is мы (we), and these are past tense plural forms.

In Russian past tense:

• masculine singular: often ends in -л
• feminine singular: often ends in -ла
• neuter singular: often ends in -ло
• plural: often ends in -ли

So:

• я ждал / ждала = I waited
• он ждал = he waited
• она ждала = she waited
• оно ждало = it waited
• мы ждали = we waited
• они ждали = they waited

Same with потратили:

• он потратил
• она потратила
• мы потратили

Here только means only, but in a sense closer to:

• all we did was...
• merely
• nothing but

So:

• и только потратили время впустую

means something like:

• and only ended up wasting time
• and all we did was waste time

It emphasizes that the waiting had no useful result.

Yes. Russian word order is more flexible than English word order, though the basic meaning would stay the same.

The original order is:

• Мы напрасно ждали автобус у другого входа и только потратили время впустую.

This sounds natural and neutral.

But Russian can move parts around for emphasis, for example:

• Мы ждали автобус у другого входа напрасно...
• У другого входа мы напрасно ждали автобус...

These versions shift the focus slightly:

• putting напрасно later can stress the uselessness
• putting у другого входа first can stress the location

So yes, the order can change, but the original sentence is a very normal way to say it.

It can mean either, depending on context, but very often it means the other entrance.

The adjective другой can mean:

• other
• another
• different

In this sentence, since there is likely a known place with more than one entrance, у другого входа most naturally means:

• at the other entrance

If the context were less specific, it could also be understood as:

• at a different entrance

So the exact English translation depends on the situation.

Because Russian does not have articles like a and the.

So Russian simply says:

• автобус
• вход

and the listener understands from context whether English would need:

• a bus / the bus
• an entrance / the entrance / the other entrance

In this sentence, context strongly suggests:

• the bus
• the other entrance

This is very normal in Russian: definiteness is usually understood from the situation rather than shown with an article.

Yes, it could.

Russian often drops subject pronouns when they are clear from the verb form and context. So this sentence could also be:

• Напрасно ждали автобус у другого входа и только потратили время впустую.

That still clearly implies we, because the past tense plural verbs ждали and потратили show that the subject is plural.

Including Мы makes the subject more explicit and can add a slight sense of emphasis:

• We waited in vain...

So both are possible.