Investigate city is the cheapest and safest way to get the same info later. You can also use it after another civ has built an embassy in your capital.
I don't believe there is any way to know an opponent's research project or progress for sure. However there is an article somewhere near here that helps to predict which techs they are likely to research next, and if you keep up a regular visiting schedule for trade talks and don't get left too far behind you can get a feel for when they complete one project and start another. You would have to keep up to within one tech of the competition, otherwise, for example, if you don't have Writing you won't know when they complete Map Reading as it won't appear in their trading list.
If you can get a feel for when they start a project you will still have to guess what rate they will research it at, based on experience, difficulty level and your assessment of their research budget.
