Dear Ms. Demeanor.
I have been wanting to buy my neighbor's apartment and combine it with mine for what feels like eternity! I hear a rumor she is thinking about retiring to Florida. How can I ask her without seeming rude or like I'm trying to push her out (even though I am)?
Your approach depends entirely on your relationship with her. If you two have a warm relationship, invite her over for cake and coffee and let her know you are interested when she is ready.
If your interactions with her are limited and/or formal, a letter is best. The letter should say that you have heard that she may be moving and, though you are sorry to see her go, you would be very interested in purchasing the apartment should she choose to sell.
If your relationship has had its ups and downs, tip the doorman to tip off the resident (and the resident's real estate agent), relying on the wisdom of "a bird in the hand...". Be genuine and keep your eye on the prize.
To combine is divine,