Don't take Ibuprofen.
(I was prescribed it in the last few weeks of pregnancy. Result - one beautiful baby with a heart murmur - hence the rules now).
Physical treatment - such as shiatsu/osteopathy with somebody well versed in pregnancy contraindications, or, preferably/if you can't afford that, physio - combined with
attention to whether your head is supported properly and not flopping over or being pushed up at an angle whilst you're in bed, even trying a foam shaped pillow,
work on your posture (your centre of gravity might be out from the extra weight/poor core muscle tone and it's possible that you're pushing your head/chin out instead of tucking it in and lengthening your spine)
and gentle mobilisation stretches will be much safer. A physio would be able to help plan safe exercise to do in pregnancy, but might not do any hands on work, whereas a shiatsu practitioner or Osteopath if they're happy to work on you,
Cold packs are brilliant, if somewhat uncomfortable. Paracetamol is, as far as I know (check with your GP) safe in pregnancy - have two with a coffee or cocoa-cola at the first hint of heaviness. Tiger Balm smells horrid, but if your skin (or stomach) can tolerate it, it's very effective.
If you can still lie on your front, try laying flat with your forehead down and your arms by your sides. Can you lift your head at all without having to use your arms? If not, you've got very weak neck muscles, so they're going to struggle to keep your head up, especially as pregnancy will loosen everything. Those gentle exercises as advised by a physio will help increase your strength safely.
You could also look at pregnancy yoga. Even if it doesn't sort out your neck/shoulder completely, it'll be beneficial for you in terms of your entire body and mind.
I continued Osteopathy sessions throughout my first pregnancy. That woman had the ability to stop a migraine in its tracks and my yoga sessions left me feeling slightly high, I was so peaceful after them.
I still use my neck pillow now, about a million years later. It was horrible the first night, not much better the second - but on the morning of the 3rd day after the 2nd night, I woke up, moved ever so slightly, my neck cracked louder than I had ever heard it do before or since and I've been pretty much fine ever since. I won't change it just in case the pain comes back - it's just the cheap IKEA one that has a low edge and a higher edge and the only way I'm being separated from it is if it's torn from my cold, dead hands.