HI Michelle, good afternoon from Port Moresby, I hope you are excited about your PNG trip?
I know a few cheaper places but I would however agree with Steve recommending Ela Beach Hotel considering you're a female and travelling alone. Perhaps I might suggest you look at City Service Apartment(New Hotel) and it also have a boutique hotel too just near the market if you like market and also near the football stadium which you don't have to pay to watch a great footy if you love that. Staffs are friendly to. Apart from the list Steve gave you should consider that too and see I trust whichever you choose you will have a great time.
I have a few Taxi Drivers and are very friendly. I hooked up couple of tourist friends with them and they are very happy for the service. If interested let me know and we can keep in touch via my message box or if you need to have my contact I am more than pleased to help out. If you prefer other wise for transportation, Here's the taxi Company I trust Comfort Taxi, City Loop, Tulait tulait. Always seek advice from your hotels reception if you are travelling alone they sometimes have few taxi trusted drivers so you should be ok.
Another thing, if you are thinking of travelling out of the City to Adventure Park, Sogeri for a great sight, I can also try to help . My niece have and her bf have an open back, you might wanna experience a local llifestyle? We'll be happy to help out. But if you get to meet Steve and his wife then you are in a really good hands.
I wish you all the best.
Kindest Regards,
AJM