I corrected your sentences but then I rewrote them to make them sound better.  So the first correction is to show corrections in grammar.  Then second correction is to make the sentences sound better.  

January 5, 2015

I got up early in the morning today like always. 
Then I ate breakfast and got dressed.  I headed out to work at 7:10.  It would be better to say " I left for work at 7:10am.")  

It would be better to say:

This morning I woke up early as always.  Then I eat breakfast, got dressed, and headed out to the bus stop at 7:10.

The bus came at 7:20 and I got off the bus at 7:45.  From there, I walked to work and arrived at work at exactly 8 O'clock.

It is better to say:  

At 7:20 I took the ( you should write the town it took you to.)  From there, I walked to work and arrived at exactly 8 O'clock.

I made coffee for myself and started working.  Then it was all work and work and work until 1 PM.

At one O'clock, like always, I had an hour to rest and eat lunch.

It is better to say:

When I got work, I made coffee and started to work.  Then it was all work until 1p.m., when I took my hour lunch (break) as usual.  

Today wasn't a busy day but I worked on a program that I hadn't enough knowledge about it and it made me very tired.   

Today was not a busy day, but I did work on a challenging program, which I did not have much knowledge about.  This made me tired.  I hope to slove the problem/issue tomorrow.  OR (" I hope to resovle the problem/ issue tomorrow."

I hope to solve my issue tomorrow :)