From BYO an article on how to measure IBUs of beer. All you need is a UV/VIS spectrophotometer. Most high school and all college chem and bio lab will have several that they might let you use in exchange for a six pack.
The formula is IBUs = 50 * (the absorbence at 275nm)
The article isn't clear on the procedure though....it says a "simple solvent extraction is used", and since it's talking IBUs I'm guessing it's a liquid-liquid extraction. The details aren't specified however. I think it is more complicated then just a few drops of beer in your UV/VIS spectrometer though.
Hmm, some Googling turned this procedure up from some chem lab:
Student sheet: Bitterness of beer
Scope
The determination of the bitter substances in beer, which are mainly iso-R-acids (with a five member ring) originating from R-acids present in hops.
Principle
Bitterness is an essential quality parameter in modern breweries, and analysis of bitterness in beer and wort is conducted as a routine throughout the brewing industry. The traditional and international recommended analysis of bitterness in beer, in international bitter units (IBU), is carried out by a spectrophotometric measurement at 275 nm of an acidic solvent extract of beer. Acidified beer is
extracted with 2,2,4-trimethyl pentane. The bitter substances dissolve in the organic phase. After centrifugation, the absorbance of the 2,2,4-trimethyl pentane phase is measured at 275 nm against pure 2,2,4-trimethyl pentane.
Reagent solutions
hydrochloric acid, 6 mol/L
Preparation
Sample
Degas the beer samples by gently stirring with a magnetic stirrer on low speed. Degassing agents and filtration may decrease the bitterness and is therefore not recommended. Plastic containers must not be used.
Iso-R-acids adsorb onto glass. Rinse thoroughly cleaned and/or new glassware first with an 2,2,4-trimethyl pentane beer extract before use.
Procedure
Pipette 10.0 mL degassed beer into a 35 mL centrifuge tube or a 50 mL conical flask. Add 0.5 mL HCl (6 mol/L) and 20 mL 2,2,4-trimethyl pentane. Place 2 to 3 glass balls in the tube. Screw a cap on the centrifuge tube or stopper the flask, to get a solvent tight seal. Shake the centrifuge tubes for 15 minutes at 20 ± 1 °C in a rotary shaker set at 130 ± 5 rpm or use a platform shaker and shake 20 ± 1 °C until a maximum extraction has been achieved, or shake by hand. Measure the absorbance of the 2,2,4-trimethyl pentane phase at 275 nm at intervals during shaking until no further increase in absorbance is observed, to establish the time of shaking.
Allow the emulsion to settle, centrifuge the tubes for 3 minutes at 3000 rpm (or use a separation funnel).
Measure the absorbance of the 2,2,4-trimethyl pentane phase in a 10 mm cuvette at 275 nm, using pure 2,2,4-trimethyl pentane as a reference. Before measuring, the cuvette must be washed with a part of the 2,2,4-trimethyl pentane phase. Record the sample absorbance in a few seconds. Absorbance measurement should be made within 20 minutes after centrifugation (or separation in the funnel) because further delay will affect the results.
Calculation
Bitterness (IBU) = 50 × A275
where A275 is the absorbance at 275 nm measured against 2,2,4-trimethyl pentane as a reference.
Source (Warning: PDF)