Hi Full Monty,
When to visit Muscat would depend largely on what exactly you wish to see and do.
For example, if you are a museum lover and a culture vulture, Muscat has over 20 different museums. But sadly they are open only during weekdays and during official business hours only. And are closed on weekends and public holidays (yes, on all public holidays all museums are close - which should ideally be kept open so that visitors can visit them !).
As you may probably be aware, Muscat is not a nightlife destination like say, for example, Dubai is. Most establishments down their shutters by 9ish in the evenings. Malls might be kept open a little longer.
Just looking around the city's landmarks would be a good thing for you to do - like take a walk on the Muttrah Corniche, check out the famous Muttrah Souq, see that famous Riyam Park incense burner tower overlooking the bay of Muscat, visit the Grand Mosque, drive down to the Oman Dive Centre / Marina Bandar Al Rowdha - for the scenic route and for a visit to the marina there. Go early in the morning and you could go for a Dolphin Watching trip from there (am not sure if they have charters on weekdays though). Visit the Royal Opera House Muscat. Once you plan your dates, book your tickets online and catch some exciting show there when you are here. A visit to the Royal Opera House Muscat should be a 'must do' on your itinerary.
Muscat has some beautiful beaches. Walk leisurely along the 2 km Shatti beachfront. There are some nice eateries for you to enjoy a pleasant evening. But I think as it is already turning warm, March may be a bit sultry already. You can also visit some nightclubs which can be found around the Shatti beachfront neighbourhood - like in Crowne Plaza, Ramee International, etc.
Day trips can be done to Nizwa where you can visit the famous Nizwa Fort and see the Nizwa Souq. Just past Nizwa is Bahla which has its own fort too. A little further ahead is Jabrin Castle. This would give you an idea about Oman's ancient history, culture and traditions.
To go through a travel agency as a solo traveller might work out rather expensive. Check it out anyway. There are some who cater exclusively to in-bound tourists like yourself. Do a google for such travel agents online and you will get their names. But be warned. Though they may be reliable, they would rip you off ! If you drive, come with an international driving licence, hire a rent-a-car and explore by yourself. You can stop wherever you wish and ask people for directions and you will always find someone to offer you directions, guidance and assistance. Also most hotels offer concierge services and you could plan daily trips in consultations with them.
Muscat is a relatively easy city to move around. And you should be just fine. But as any solo traveller should be, always be alert.
Have a pleasant trip and good luck to you !