Yes, there are many free software options available for recording solo acoustic and voice on your computer.
Audacity - Audacity is a free and open-source audio editing software that allows you to record, edit and mix your audio tracks. It's a popular choice for beginners and offers many features for editing and enhancing your recordings.
GarageBand - If you have a Mac, GarageBand comes pre-installed and is a great option for recording acoustic and vocal tracks. It's intuitive to use and offers many built-in effects and tools for editing your recordings.
WavePad - WavePad is a free audio editing software that allows you to record and edit your audio tracks. It offers many features for enhancing your recordings, such as noise reduction and EQ adjustments.
Ocenaudio - Ocenaudio is a free and easy-to-use audio editing software that offers many features for recording and editing your tracks. It's a great option for beginners and offers many built-in effects and tools for enhancing your recordings.
LMMS - LMMS is a free and open-source digital audio workstation that allows you to create and edit your own music tracks. It offers many features for recording and editing your audio tracks and is a great option if you want to create more complex music compositions.
All of these software options should be able to work well with your current cheap microphone and help you create high-quality recordings to upload to YouTube.