Can I solder a wire to a PCB without a soldering iron?
Soldering makes a strong bond between the two metals with resistance that is as small as possible. I am unaware of any other bonding method that works equally well, especially if you consider its mechanical properties. A glue-type solution might hold for a while but might fall off later on because of vibrations.

In addition, any glue will go between the two metals and will not form a continuous conductor unless it is a conductor itself. There are some conductive adhesives but I expect them to cost more than a cheap soldering iron and I wouldn't be able to tell you how well it would hold without having tried it. Also, as the linked wiki says, it is consisted 80% of conducting material so I would expect some added resistance which in turn could cause extra heat.

If you decide that you're going to have an attempt to solder it, you should become familiar with the soldering iron first. A broken piece of old electronics you don't need could make a good practice board.

Keep in mind that there are two (more actually) types of soler - leaded and unleaded. You can't tell just by looking but the later melts at a higher temperature.

There are two things I can't stress enough about soldering: surface cleanliness and stability during soldering. You can achieve 'cleanliness' by cleaning the surface with isopropyl alcohol or acetone. Some flux will also help remove oxides from both surfaces. Flux is applied before solder. Most solder has some flux in its core these days but you can't go wrong if you apply some to each surface before applying solder. Finally regarding flux, many types require cleaning afterwards (usually with isopropyl alcohol or acetone) otherwise they will corrode the solder joint after some time.

Stability will take a little bit of practice but if you hold the pcb down and have something hold the wire in place (like tape or blue tack) it's actually not that hard.

One last tip, always pre-tin (apply solder to) both parts before joining them and have the soldering iron tip wet with solder when you try it. This helps with heat transfer and make the whole thing faster which in turn means less stress on the PCB.

There are some really good videos online that can demonstrate all these points in a nicer way than text. This is an example that I consider a very good one.

If this is valuable to you and you don't feel comfortable with a soldering iron, you should take it to a technician. This shouldn't take him/her more than a few of minutes.