Question: I will be away for a couple of days in Elul without a minyan. Do I need to blow the shofar for myself?
Answer: The Tur (OC 581) gives a couple of reasons for the minhag to blow the shofar throughout the month of Elul. According to the Pirkei Derebbi Eliezer (46), the shofar was blown in the camp when Moshe went up Har Sinai on Rosh Chodesh Elul to get the second set of luchos. Chazal instituted that we blow the shofar in commemoration of this on Rosh Chodesh. The Tur adds that the minhag is to blow throughout Elul to stimulate us to do teshuva (See Rambam, Teshuva 3:4) as well as to confuse Satan.
R’ Ephraim Greenblatt (Rivevos Ephraim 1:394; 8:523:4) quotes R’ Avraham Yitzchak Kook (Mitzvas Ra’ayah 581:1) as giving another reason. Just as the halacha is that one must start learning the halachos of Yom Tov thirty days in advance (Pesachim 6a), so too, we need to begin practise blowing the shofar thirty days before Rosh Hashana.
Depending on the reason for blowing, R’ Greenblatt writes that there is a debate as to whether one davening alone should make the effort to hear the shofar himself, though he concludes that it is unnecessary.
R’ Eliezer Waldenberg (Tzitz Eliezer 12:48) writes similarly that the various pesukim that talk of the shofar being blown refer it to being blown in the camp and in the city. Hence, one davening alone does not need to blow the shofar.
In conclusion, there is no reason to blow the shofar during Elul when not with a minyan.