hi,
no way: 4051 is not ISP chip, it has neither boot-loader nor ISP hardware. This chip requires parallel programming with external 12V. Usage of external hardware takes board space, money and additional framework. And is it for what? Only 1.000 erase/write cycles are possible max.
For experiment purposes, well take one more MCU or PLD which:
- receives data to be stored;
- resets 4051;
- puts it into programming mode;
- full erase 4051;
- write its program back (yeah! :)
- adds stored data at the end;
- take off programming conditions;
- restart a 4051.
Another way: buy T89C5115 - low pin count MCU with 16kb ISP Flash (via bootloader) and 2kb of EEPROM. Both memory spaces may be software programmed on-the-fly about 100.000 times.

Always try to break the rules!
I have heard that AT89C4051 has not 4kb but 8kb of program memory. Try to find that "secret area", now! (just a joke :)