How is 以 used in 时间给勤奋者以荣誉，给懒汉以耻辱?
Here's what Paul Rouzer has to say on 以 in his New Practical Primer of Literary Chinese:


  1.5 以: This is one of the most frequently used characters in literary Chinese, and it has a very wide application. It was originally a verb meaning “to take,” “to use.” Eventually, it started to be used in combination with other verbs (what English language scholars often call a coverb). It serves the same purpose as English prepositions like “with,” “by means of,” and “through.” It usually comes before the main verb, and it is followed by its own special object:
  
  天以禍報爲不善者。Heaven repays with disaster the person who does evil.
  
  In this example, 報 is the main verb, and 爲不善者 is its object. 以
  is the coverb, and 禍 is the object of the coverb.
  
  Other examples:
  
  Identify the object of the coverb in each case: 
  
  聖人以德導小人。 The wise person leads the petty person with virtue. 
  
  自禁者以德導欲。 The person who restrains herself leads her desires with virtue.
  
  1.6 Shifting the coverb position: However, if a writer wants to put special emphasis on the thing that gets used (as in text #3), he or she will place the coverb and its object after the main verb:
  
  小人導心以耳目: The petty person leads his heart with his ears and eyes.
  
  Another, more awkward way of expressing the same sense: “It is with his ears and
  eyes that the petty person leads his heart."
  
  知己者報人以善: It is with good that the one who knows herself rewards others.
  
  小人報天以怨: It is with resentment that the petty person rewards Heaven.


Based on point 1.5, a literal translation of 時間給勤奮者以榮譽，給懶漢以恥辱 might be:

時間  給       勤奮者       以   榮譽  ，     給       懶漢     以    恥辱
Time provides the diligent with honour, and provides the lazy with disgrace.


Additionally, according to point 1.6, when the coverb (here 以) is placed after the main verb (here 給), it puts more emphasis on the thing after the coverb (here 榮譽 ‘honour’ and 恥辱 ‘disgrace’). Therefore, an even more literal translation (though a very awkward one) could be:

It is with honour that time provides the diligent, and it is with disgrace that time provides the lazy.

That should explain the use of 以, which in this case basically means ‘with’. Of course, a natural-sounding translation cannot be as literal as the ones above.

Note: 以 can be translated as ‘with’ in this particular context. It doesn't mean that ‘with’ is always a correct translation if 以. In particular, 以 does not mean ‘together with’, and is therefore different from words such as 和，与，跟， 一起 and so on. As we can see in the above quote, 以 may in other context be expressed with ‘take’, ‘use’, ‘by means of’ and ‘through’.