Here is how to get from a Disney Resort to Universal the cheapest way:
1: Take WDW free transportation to DownTown Disney
2: Take Lynx bus 50 from there to Sea World. 20 minutes journey time; frequency every 30 minutes.
3: Take Lynx bus 8 from Sea World to Sand Lake Rd/I-Drive intersection. 20-25 minutes journey time; frequency 10-15 minutes
4: Take Lynx bus 21 from there right to to Universal Studios. 20 minutes journey time; frequency every 30 minutes
Cost is $1.50 per person each way on Lynx regardless of distance. Pay just once as you board the first bus, ask for a "transfer" and then use this when you board the next buses. The whole trip could take up to 2 hours - maybe you will meet somebody at your accommodation who drives and would give you a lift, otherwise maybe stop at Seaworld for a while and continue from there.
P.S. All the buses mentioned in the above post do run late-ish. In this particular case, the limiting factor is bus 50 on your return journey - the last one leaves SeaWorld for Disney at 10.10 pm, so you can comfortably stay at Universal until probably 9ish.