For someone who davens without a minyan, there is no obligation to say Piyutim, but it is certainly appropriate to do so. Most Piyutim are said in entirety by both the Chazzan and congregation, and that is how the individual would recite them as well. [Although parts of Piyutim are often marked as Chazzan and Kahal in the Machzor, this is only in order to highlight which parts the Chazzan says aloud.]
Exceptions: A non-Chazan does not say the paragraph of Misod that appears at the beginning of the Chazans repetition, and in Shacharit he also does not say the paragraph that follows Misod. When saying them alone, the piyutim should not be recited within the Amidah. Reading them after Shacharit and Musaf respectively, is fine. You may also repeat Aleinu as part of your personal reliving the Repetition of Musaf.
The paragraphs of Hayom Harat Olam may also be said alone, followed by the appropriate passage beginning Areshet Sefateinu. But don't recite the above paragraphs during the first series of Shofar blowing (of 30 sounds; aka Tekiot diMeyushav).
When one davens alone, it is appropriate to read Kriat Hatorah and Haftorah in private between Shacharit and Musaf. If someone is davening without a Minyan, he should not hear Shofar or recite Musaf during the first quarter of the day (after Sof Zman Kriat Shma), unless there is no other option. When davening alone, the Shofar should be blown only before Musaf but not during the Amidah and the person blowing should consider the other Minyanim outdoors in earshot.
You may need for someone else to blow Shofar for you. Where possible, this should be done after the first quarter of the day (after Sof Zman Kriat Shma) has passed. Some have the custom not to eat or drink before Shofar blowing, however if the Shofar blower cannot come until late in the day, those who do not eat prior to Shofar may eat. After he concludes the Amidah, he may recite the Piyuttim normally recited during the Chazzan’s repetition, but there is no obligation for him to do so.
