A 1 month old can get food from a few places: the breast, a supplemental nursing system, or a bottle. By 4 months you could potentially introduce cup feeding or spoon feeding of formula or breastmilk, but not all children can do it. However during the newborn phase your child is designed to get comfort from sucking on the breast or bottle. It's OK.
Bottle addiction varies from child to child. Some children, indeed, do not want to give up their beloved bottles when they are young toddlers. Others switch easiliy to a transitional sippy cup like this one or this one or a straw cup and eventually to a tumbler.
One thing that may be helpful in preventing a bottle addiction is to provide your child with other comfort objects: a pacifier now and a soft lovely at an appropriate age, for example. That way when the time comes to remove the bottle, my pediatrician said at 15 months but the recommendations vary, your toddler is not left without a source of comfort.
When looking at how to transition my child from a bottle to a sippy cup I discovered that most parents use two approaches.
Take the bottle away cold turkey and only offer sippy cups. Most young toddlers are upset with this change for 1-2 days and then forget about it. Long-term memory does not fully develop until speech develops so the transition from bottle to sippy using a cold turkey method will be easier earlier. We successfully used this method with my child at 13 months. I simply moved her from a bottle (which we had given her after I began weaning her from breastfeeding at 12 months) to a soft-spout sippy cup. She did not even protest. We made sure to make the transition during a time when she seemed to be developmentally stable.
Wait until the toddler is ready to give up the bottle. Many pediatricians and pediatric dentists would not approve of this practice, but many parents use it. If you do use it, make sure that your child is not getting a bottle after tooth brushing beyond when your pediatrician recommends. As your child ages limit bottle use to meal times. Under this method many toddlers will eventually express a willingness to switch, or you can encourage it by using one of the many tricks for older toddlers such as switching bottles with a new toy.