I'd like to suggest an exercise which I believe will give you a very strong sense of time within 2-3 months. The exercise is quite enjoyable too. I'll first describe it and then explain the logic behind it.
Basically, buy a a pair of drumsticks and practice keeping time with them while you listen to music.
You can tap on any surface, but if you can afford to get a rubber drummer training pad that'll be even more enjoyable.
At first practice playing all the downbeats. Aim at playing so precisely that the sound of your tapping disappears into the music's downbeats.
Keep doing that until you can tap very precisely along with ordinary songs from beginning to end. at first you'll get distracted, confused, etc. but just keep going and your brain will get it, you'll eventually totally master this.
When the basic tapping becomes too easy, start to tap only on the upbeats. (Only on the "and" portions). It will be tricky at first, but again, keep practicing, it'll get easier and easier and eventually effortless.
When that becomes easy too, add some variations. For example, alternate one bar where you tap the down beats with one bar where you tap only the upbeats, or any other combination you fancy.
This can be very enjoyable too, because you can do it while listening to the music you like anyway. In fact, it will give you a better understanding of the music you hear, too.
This exercise leads to internalizing the sense of the beat to a very high degree.
And the reason is that your physical movement (moving the stick) and the sharp sound of each "tick" of the stick become deeply ingrained in your brain, and will go on subconsciously whenever you listen or play.
Notice that the same internalization will NOT happen if you only listen to the music passively -- you will have to practice coordinating a well defined physical movement (your arm's swinging the stick) with the beat of the music you hear. And you will also need to feed a very precise reference -- the sound of the stick hitting a surface -- to your brain as well. For example, if you only practiced by waving your arms like the director of an orchestra, you will not get the same benefit as using a drumstick, not even close.
My personal experience, when I figured out I could do this exercise and started doing it during long commutes (usually tapping the padding of the next seat on an empty bus I used to take daily) is that it made a dramatic impact in the quality of my sense of time, groove, and quality of playing.