Should "Project Manager" be capitalized?

The Guardian style guide says:

jobs all lc, eg prime minister, US secretary of state, chief rabbi, editor of the Guardian.

titles cap up titles, but not job description, eg President Barack Obama (but the US president, Barack Obama, and Obama on subsequent mention); the Duke of Westminster (the duke at second mention); Pope Benedict XVI but the pope.

Project manager is a job not a title, so should be lowercase in both examples:

"The project manager will evaluate the alternatives and offer a solution."

"A project manager is the person who manages the budget and deliverable for a project."

Agreed on both counts regarding the examples in the OP. That is, job titles should be capitalized when they are taking the place of a single person (or otherwise acting as a name of an entity). However, there is no need to capitalize in other cases, and it would be strange if capitalized in the second example, unless it were in a company operations manual, for example.

A similar example is when you capitalize "dad" in the sentence:

I'm going fishing with Dad tomorrow.

However you don't capitalize it in the following version:

I'm going fishing with my dad tomorrow.

The former replaces a name, where the latter simply states the relationship of the person.

As Lee answered, when you replace a name with a pronoun, you capitalize it. However your first example is using it as an improper noun and it should simply read:

"The project manager..."

The only job titles I can think of to capitalize are honorifics that are included with names - like royalty. I would for instance capitalize:

Her Royal Highness, Queen Victoria and Professor Moriarty