The Ryzen APUs are a great choice for the budget. They'll handle what you need really well. Similar to the build from the video you linked but faster RAM (highly beneficial on Ryzen APUs) and higher quality Motherboard with far more expansion freedom (4 RAM slots for example).
PCPartPicker part list:
https://pcpartpicker.com/list/T4TyHh
Price breakdown by merchant: https://pcpartpicker.com/list/T4TyHh/by_merchant/
CPU: AMD - Ryzen 3 2200G 3.5GHz Quad-Core Processor ($98.79 @ OutletPC)
Motherboard: Gigabyte - B450M DS3H Micro ATX AM4 Motherboard ($69.99 @ Amazon)
Memory: Patriot - Viper 4 8GB (2 x 4GB) DDR4-3000 Memory ($80.98 @ Newegg)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($43.90 @ OutletPC)
Case: Cougar - MX330 ATX Mid Tower Case ($34.99 @ SuperBiiz)
Power Supply: Corsair - CX (2017) 550W 80+ Bronze Certified ATX Power Supply ($32.98 @ Newegg)
Total: $361.63
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-08-30 23:23 EDT-0400
If you're happy to spend the extra for an SSD boot drive:
PCPartPicker part list:
https://pcpartpicker.com/list/3psGhy
Price breakdown by merchant: https://pcpartpicker.com/list/3psGhy/by_merchant/
CPU: AMD - Ryzen 3 2200G 3.5GHz Quad-Core Processor ($98.79 @ OutletPC)
Motherboard: Gigabyte - B450M DS3H Micro ATX AM4 Motherboard ($69.99 @ Amazon)
Memory: Patriot - Viper 4 8GB (2 x 4GB) DDR4-3000 Memory ($80.98 @ Newegg)
Storage: ADATA - Ultimate SU650 120GB 2.5" Solid State Drive ($29.89 @ OutletPC)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($43.90 @ OutletPC)
Case: Cougar - MX330 ATX Mid Tower Case ($34.99 @ SuperBiiz)
Power Supply: Corsair - CX (2017) 550W 80+ Bronze Certified ATX Power Supply ($32.98 @ Newegg)
Total: $391.52
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-08-30 23:25 EDT-0400
Hope this helps!