Can I upgrade my ESP8266 firmware by connecting the pins in Arduino Uno, or do I still need a USB UART to Serial?
Looking at the schematic of arduino UNO here you can see that Pin 0 and Pin 1 are connected to the same line of RX TX of the ATMEGA8U2. So in theory (and assuming the ATMEGA8U2 behaves like any usb to uart IC) if you remove or disable the ATMEGA328P, your arduino should behave like any cheap usb to serial. The way you can "disable" ATMEGA328P is by setting it in constant reset mode as explained here
By manually reseting and setting the esp into flashing mode, you should be able to upload the new firmware to it from arduino IDE for example by selecting the ESP device.
Two notes on this
1 - The TX line of the ATMEGA8U2 is the marked RX on arduino Uno board and RX line of the ATMEGA8U2 is the marked to TX on arduino Uno board. That means your ESP RX should connect to pin 0 on arduino and TX of ESP should connect to pin 1.
2 - You might need voltage level conversion on pin 0 as ESP is a 3.3V device although in schematic there is 1K resistor in that line