First, the ARM7TDMI code is used in many different vendor's chips. Different vendors/packages have some differences in number of I/O pins and other peripherals available. So, which vendor/mfgr and chip are you planning to use? You will need to use and configure a number of the GPIO pins of the particular chip you wish to use to handle the addressing, control lines, and databus duties to interface with the ADC0808. Pretty much all of the GPIO on these devices is 5V tolerant, so you won't have to use an intermediary voltage level shifting chip - you could simply use pullup resistors to +5V where appropriate. You then would have to write software routines to read or write to the ADC0808 that would read&write the values of the affected GPIO lines correctly according to the sequencing/timing requirements on the ADC0808 datasheet. Depending on how you plan to use the ADC0808, I could see you having to use up to 14 GPIO pins on the ARM chip: 2 for Read/Write control, 3 for addressing, 8 for the 8-bit wide data bus, and 1 for the ADC0808 INTerrupt signal. This uses up quite alot of GPIO pins which could be used for other things, but if your design doesn't require other IO then that is fine. Otherwise, I'd look for a different ADC chip which uses I2C or SPI interface - you will have to write the driver code for the SPI or I2C controller, but examples of that are available on the internet, whereas you would be writing completely custom/unique code for implementing the interface to the ADC0808 (well, I am sure that many others have written that sort of thing before also, but depending upon what pins you use for what function in the interface, it would be unique, whereas the basic function of reading or writing bytes on the SPI or I2C ports on the ARM7 is well understood and much more common).
If you already have an appropriate JTAG programming/debugging pod, such as a Segger J-Link, you can obtain free trial versions of either IAR or Keil development tools which have all the features you need to write the code and program the chips. If you need a programming device as well, both of these companies have low cost introductory/evaluation kits which you could buy. It seems like you may be a student, and often there are student discounts also available.
Another development tools suite with programming and debugging hardware for the ARM chips which I also use is Rowley Crossworks, and they have some low-cost options available also for student/educational use.