Je prends le métro tous les jours pour aller au boulot.

Both are correct, but they emphasize slightly different things:

• prendre le métro (take the subway) focuses on the act of boarding or catching it.
• aller en métro (go by subway) highlights your mode of travel.
  Use prendre when you talk about the action of getting on. Use aller en when you simply describe how you travel.

French generally requires a definite article (le, la, les) before singular and plural nouns when speaking in general terms.

• Je prends le métro means “I take the subway” in general.
  If you want to stress one specific train, you could say Je prends un métro (“I catch a subway train”). But Je prends métro without an article is ungrammatical.

Tous les jours literally means “all the days,” i.e. every day.

• Tous has an -s because it agrees in gender (masculine) and number (plural) with jours.
  You could also say chaque jour for “each day,” but tous les jours is the most common everyday expression.

Pour + infinitive (here aller) expresses purpose or intention, equivalent to English “in order to.”

• Je prends le métro pour aller au boulot
  = “I take the subway in order to go to work” (i.e. that’s the reason I take it).

Boulot is informal slang for travail or job.

• In casual conversation, you’ll hear boulot a lot: “J’arrive au boulot,” “Je cherche du boulot,” etc.
• In a formal context, use travail (work) or emploi (employment/job).

In French, à + le always contracts to au before a masculine singular noun.

• le boulot is masculine, so à le boulot becomes au boulot.

Yes. Je vais au boulot en métro tous les jours is perfectly natural and means the same thing.

• Je prends le métro emphasizes the act of boarding the métro.
• Je vais en métro emphasizes the mode of transport. Both convey that you commute by subway every day.