Prepositions – ‘At the Beginning’ vs. ‘In the Beginning’

To me these two statements have different meanings.

What are the alternatives to this project?

This means "What other projects could we do instead?"

What are the alternatives for this project?

This means "In the context of this project, what choices do we have about how to solve it?

In a comment responding to the original post, Eduardo advises the poster to add a verb in front of the phrases "at a hotel" and "in a hotel" to obtain a more precise idea of which phrase is more common. I ran Ngram-based Google Books searches for three pairs of verbs + the phrase "in/at a hotel" over the period 1850–2005 First, "met at a hotel" (blue line) vs. "met in a hotel" (red line):

Second, "stayed at a hotel" (blue line) vs. "stayed in a hotel" (red line):

Third, "stopped at a hotel" (blue line) vs. "stopped in a hotel" (red line):

Clearly, the choice of verb has a considerable effect on the choice of preposition, assuming that all other factors remain effectively equal. But that's a major and probably unjustified assumption.

One major further complication involves the presence or absence of a noun following "hotel" in the phrase. In all three verb phrases that I tested, the "in" phrase was likelier than the "at" phrase to draw a following noun.

For instance, in the Google Books/Ngram results for "met at a hotel" versus "met in a hotel," in the first ten results for the 1987-1998 time period, "met in a hotel" was followed by a noun—suite, elevator, bar, room (three times), or restaurant—seven times. In contrast, "met at a hotel" for the 1938-1998 had only one following noun (suite) in the first ten results, although subsequent results offered such additions as garden brunch, swimming pool, and resort.

The phrases "stayed at a hotel" and "stayed in a hotel" were far less often followed by a noun in the Google Books results than "met at a hotel" and "met in a hotel" were. In fact, I didn't see any examples of a following noun involving "stayed at a hotel"; in contrast, "stayed in a hotel" did occasionally show up as part of "stayed in a hotel room." The Google Books search couldn't find any examples of "met at a hotel room," "met at a hotel lobby" "met at a hotel bar," or "stayed at a hotel room." It found "in" versions of all of those phrases.

These results lead me to hypothesize that, overall and in everyday usage, the phrase "in a hotel" lends itself to further narrowing by a following noun (especially room) to a significantly greater extent than the phrase "at a hotel" does—and that this phenomenon muddies the waters in any effort to determine by means of a simple test which simple phrase ending in the noun hotel ("at a hotel" or "in a hotel") is more common.