I have a bad news.

"Bad news" is not something that you can quantify/count, so you don't need "a" as a preposition.

One of the sections was under construction fromthat started this week.

There are multiple sections on a highway, so you need "s" at the end to indicate that it's plural.

To say the contruction began this week, you would say "That started", because it specifies when the construction started.

I usually take more than 30 minutes or an hour when I use it to go to my customer in morning.

You can use "usually" to say it's a habit. You also need "than" to talk about quantities. Ie "more than 30 people"

In addition, it will last until the end of this month.

you need to specify a quantity when you use "last" as a form of time ie "it will last all of this month". "it will only last a few weeks"

Who likes itconstruction?

It sounds more natural to repeat construction here so the reader understands what you're talking about

The worst month hads started.

Your story is in the present tense, so you should keep it consistent.

I have a bad news.

We almost never say "a news", we would say "a piece of news" or something along those lines. However, in this case just "news" with no article fits the best.

I often drive on a certain highways.

Given later sentences it makes sense to specify that you're talking about a particular highway.

One of the section was is under construction fromstarting this week.

Here it makes sense to express this in the present tense if it is still under construction.

It takes more 30 minutes tor an hour longer when I use it to go to my customer in morning.

In addition, it will last thise entire month.

Adding a word like "entire" or "whole" emphasises how much time you think it is.

WIs there anyone who likes it?

Adding this extra part at the start emphasises the rhetorical nature of the question.

The worst month hads started.

Present tense makes most tense since the month is still in-progress.