In this scenario is my electrician giving good advice?
Would changing the fuse box mean that I can use energy saving bulbs again?


No, you can use energy saving bulbs on a 30-year-old fuse-box.

Fuse-boxes don't contain fuses any more, they contain circuit-breakers (of varying types), so you'll find that "consumer unit" is the current name for what were once fuse-boxes.

Replacing a consumer-unit would be expensive.


  replace one switch (a dimmer that makes a buzzing sound


Most dimmers are incompatible with energy saving bulbs (CFL). Unless you really want a dimming capability, you can just have that replaced with an ordinary light switch. This should be a very low cost (a competent person should be able to do this in 10 minutes, the parts would cost less than a sandwich and a coffee).